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For reasons best known to their lawyers, Red Hat have chosen to hobble their implementation of OpenSSL. According to a releated bug, possible patent issues have led them to remove a large number of the elliptic curve parametrisations, as you can see by comparing the curves supported on Fedora 25:

[flypig@blaise ~]$ openssl ecparam -list_curves
  secp256k1 : SECG curve over a 256 bit prime field
  secp384r1 : NIST/SECG curve over a 384 bit prime field
  secp521r1 : NIST/SECG curve over a 521 bit prime field
  prime256v1: X9.62/SECG curve over a 256 bit prime field

with those supported on Ubuntu 16.04:

flypig@Owen:~$ openssl ecparam -list_curves
  secp112r1 : SECG/WTLS curve over a 112 bit prime field
  secp112r2 : SECG curve over a 112 bit prime field
  secp128r1 : SECG curve over a 128 bit prime field
  secp128r2 : SECG curve over a 128 bit prime field
  secp160k1 : SECG curve over a 160 bit prime field
  secp160r1 : SECG curve over a 160 bit prime field
  secp160r2 : SECG/WTLS curve over a 160 bit prime field
  secp192k1 : SECG curve over a 192 bit prime field
  secp224k1 : SECG curve over a 224 bit prime field
  secp224r1 : NIST/SECG curve over a 224 bit prime field
  secp256k1 : SECG curve over a 256 bit prime field
  secp384r1 : NIST/SECG curve over a 384 bit prime field
  secp521r1 : NIST/SECG curve over a 521 bit prime field
  prime192v1: NIST/X9.62/SECG curve over a 192 bit prime field
  prime192v2: X9.62 curve over a 192 bit prime field
  prime192v3: X9.62 curve over a 192 bit prime field
  prime239v1: X9.62 curve over a 239 bit prime field
  prime239v2: X9.62 curve over a 239 bit prime field
  prime239v3: X9.62 curve over a 239 bit prime field
  prime256v1: X9.62/SECG curve over a 256 bit prime field
  sect113r1 : SECG curve over a 113 bit binary field
  sect113r2 : SECG curve over a 113 bit binary field
  sect131r1 : SECG/WTLS curve over a 131 bit binary field
  sect131r2 : SECG curve over a 131 bit binary field
  sect163k1 : NIST/SECG/WTLS curve over a 163 bit binary field
  sect163r1 : SECG curve over a 163 bit binary field
  sect163r2 : NIST/SECG curve over a 163 bit binary field
  sect193r1 : SECG curve over a 193 bit binary field
  sect193r2 : SECG curve over a 193 bit binary field
  sect233k1 : NIST/SECG/WTLS curve over a 233 bit binary field
  sect233r1 : NIST/SECG/WTLS curve over a 233 bit binary field
  sect239k1 : SECG curve over a 239 bit binary field
  sect283k1 : NIST/SECG curve over a 283 bit binary field
  sect283r1 : NIST/SECG curve over a 283 bit binary field
  sect409k1 : NIST/SECG curve over a 409 bit binary field
  sect409r1 : NIST/SECG curve over a 409 bit binary field
  sect571k1 : NIST/SECG curve over a 571 bit binary field
  sect571r1 : NIST/SECG curve over a 571 bit binary field
  c2pnb163v1: X9.62 curve over a 163 bit binary field
  c2pnb163v2: X9.62 curve over a 163 bit binary field
  c2pnb163v3: X9.62 curve over a 163 bit binary field
  c2pnb176v1: X9.62 curve over a 176 bit binary field
  c2tnb191v1: X9.62 curve over a 191 bit binary field
  c2tnb191v2: X9.62 curve over a 191 bit binary field
  c2tnb191v3: X9.62 curve over a 191 bit binary field
  c2pnb208w1: X9.62 curve over a 208 bit binary field
  c2tnb239v1: X9.62 curve over a 239 bit binary field
  c2tnb239v2: X9.62 curve over a 239 bit binary field
  c2tnb239v3: X9.62 curve over a 239 bit binary field
  c2pnb272w1: X9.62 curve over a 272 bit binary field
  c2pnb304w1: X9.62 curve over a 304 bit binary field
  c2tnb359v1: X9.62 curve over a 359 bit binary field
  c2pnb368w1: X9.62 curve over a 368 bit binary field
  c2tnb431r1: X9.62 curve over a 431 bit binary field
  wap-wsg-idm-ecid-wtls1: WTLS curve over a 113 bit binary field
  wap-wsg-idm-ecid-wtls3: NIST/SECG/WTLS curve over a 163 bit binary field
  wap-wsg-idm-ecid-wtls4: SECG curve over a 113 bit binary field
  wap-wsg-idm-ecid-wtls5: X9.62 curve over a 163 bit binary field
  wap-wsg-idm-ecid-wtls6: SECG/WTLS curve over a 112 bit prime field
  wap-wsg-idm-ecid-wtls7: SECG/WTLS curve over a 160 bit prime field
  wap-wsg-idm-ecid-wtls8: WTLS curve over a 112 bit prime field
  wap-wsg-idm-ecid-wtls9: WTLS curve over a 160 bit prime field
  wap-wsg-idm-ecid-wtls10: NIST/SECG/WTLS curve over a 233 bit binary field
  wap-wsg-idm-ecid-wtls11: NIST/SECG/WTLS curve over a 233 bit binary field
  wap-wsg-idm-ecid-wtls12: WTLS curvs over a 224 bit prime field
  Oakley-EC2N-3:
    IPSec/IKE/Oakley curve #3 over a 155 bit binary field.
    Not suitable for ECDSA.
    Questionable extension field!
  Oakley-EC2N-4:
    IPSec/IKE/Oakley curve #4 over a 185 bit binary field.
    Not suitable for ECDSA.
    Questionable extension field!
  brainpoolP160r1: RFC 5639 curve over a 160 bit prime field
  brainpoolP160t1: RFC 5639 curve over a 160 bit prime field
  brainpoolP192r1: RFC 5639 curve over a 192 bit prime field
  brainpoolP192t1: RFC 5639 curve over a 192 bit prime field
  brainpoolP224r1: RFC 5639 curve over a 224 bit prime field
  brainpoolP224t1: RFC 5639 curve over a 224 bit prime field
  brainpoolP256r1: RFC 5639 curve over a 256 bit prime field
  brainpoolP256t1: RFC 5639 curve over a 256 bit prime field
  brainpoolP320r1: RFC 5639 curve over a 320 bit prime field
  brainpoolP320t1: RFC 5639 curve over a 320 bit prime field
  brainpoolP384r1: RFC 5639 curve over a 384 bit prime field
  brainpoolP384t1: RFC 5639 curve over a 384 bit prime field
  brainpoolP512r1: RFC 5639 curve over a 512 bit prime field
  brainpoolP512t1: RFC 5639 curve over a 512 bit prime field

I only discovered this when trying to build a libpico rpm. The missing curves cause particular problems for Pico, because we use prime192v1 for our implementation of the Sigma-I protocol. Getting around this is awkward, since we don’t have a crypto-negotiation step (maybe there’s a lesson there, although protocol negotiation is also a source of vulnerabilities).
There’s already a bug report covering the missing covers, but given that the situation has persisted since at least 2007 and remains unresolved, it seems unlikely Red Hat’s lawyers will relent any time soon. They’ve added the 256-bit prime field version since this was licensed by the NSA, but the others remain AWOL.
Wikipedia shows the various patents expiring around 2020. Until then, one way to address the problem is to build yourself your own OpenSSL RPM without all of the disabled code. Daniel Pocock produced a nice tutorial back in 2013, but this was for Fedora 19 and OpenSSL 1.0.1e. Things have now moved on and his patch no longer works correctly, so I’ve updated his steps to cover Fedora 25.
Check out my blog post about it if you want to code along.

For most people it makes sense to use the latest (at time of writing) 1.0.2k version of OpenSSL on Fedora 25 (in which case, see my other blog post). However, if for some reason you need a slightly earlier build (version 1.0.2j to be precise), then you can switch out the middle part of the process I wrote about for 1.0.2k with the following set of commands.

# Install the fedora RPM with all the standard Red Hat patches
cd ~/rpmbuild/SRPMS
wget http://dl.fedoraproject.org/pub/fedora/linux/releases/25/Everything/source/tree/Packages/o/openssl-1.0.2j-1.fc25.src.rpm
rpm -i openssl-1.0.2j-1.fc25.src.rpm
# Install the stock OpenSSL source which doesn’t have the ECC code removed
cd ../SOURCES
wget https://www.openssl.org/source/old/1.0.2/openssl-1.0.2j.tar.gz
# Patch the spec file to avoid all of the nasty ECC-destroying patches
cd ../SPECS
wget http://www.flypig.co.uk/dnload/dnload/pico/openssl.spec.ec-1.0.2j.patch
patch -p0 < openssl.spec.ec-1.0.2j.patch
# And build
rpmbuild -bb openssl.spec

And to install the resulting RPMs:

cd ~/rpmbuild/RPMS/$(uname -i)
rpm -Uvh --force openssl-1.0.2j*rpm openssl-devel-1.0.2j*rpm openssl-libs-1.0.2j*rpm

I’m not sure why you might want to use 1.0.2j over 1.0.2k, but since I already had the patch lying around, it seemed sensible to make it available.
 

Fedora’s OpenSSL build is actually a cut-down version with many of the elliptic curve features removed due to patent concerns. These are available in stock OpenSSL and in other distros such as Ubuntu, so it’s a pain they’re not available in Fedora. Daniel Pocock provided a nice tutorial on how to build an RPM that restores the functionality, but it’s a bit old now (Fedora 19, 2013) and generated errors when I tried to follow it more recently. Here’s an updated process that’ll work for OpenSSL 1.0.2k on Fedora 26.
Prepare the system
Remove the existing openssl-devel package and install the dependencies needed to build a new one. These all have to be done as root (e.g. by adding sudo to the front of them).

rpm -e openssl-devel
dnf install rpm-build krb5-devel zlib-devel gcc gmp-devel \ 
  libcurl-devel openldap-devel NetworkManager-devel \
  NetworkManager-glib-devel sqlite-devel lksctp-tools-devel \
  perl-generators rpmdevtools

Set up an rpmbuild environment
If you don’t already have one. Something like this should do the trick.

rpmdev-setuptree

Obtain the packages and build
The following will download the sources and apply a patch to reinstate the ECC functionality. This is broadly the same as Daniel's, but with more recent package links and an updated patch to work with them.

# Install the fedora RPM with all the standard Red Hat patches
cd ~/rpmbuild/SRPMS
wget http://dl.fedoraproject.org/pub/fedora/linux/updates/25/SRPMS/o/openssl-1.0.2k-1.fc25.src.rpm
rpm -i openssl-1.0.2k-1.fc25.src.rpm
# Install the stock OpenSSL source which doesn’t have the ECC code removed
cd ../SOURCES
wget https://www.openssl.org/source/openssl-1.0.2k.tar.gz
# Patch the spec file to avoid all of the nasty ECC-destroying patches
cd ../SPECS
wget http://www.flypig.co.uk/dnload/dnload/pico/openssl.spec.ec-1.0.2k.patch
patch -p0 < openssl.spec.ec-1.0.2k.patch
# And build
rpmbuild -bb openssl.spec

Install the OpenSSL packages

cd ~/rpmbuild/RPMS/$(uname -i)
rpm -Uvh --force openssl-1.0.2k*rpm openssl-devel-1.0.2k*rpm openssl-libs-1.0.2k*rpm

Once this has completed, your ECC functionality should be restored. You can check by entering

openssl ecparam -list_curves

to list the curves your currently installed package supports. That should be it. In case you want to use the slightly older 1.0.2j version of OpenSSL, you can follow my separate post on the topic.