| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 14.1-a1 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{36} \cdot 7^{14} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$1.750834270$ |
1.101763183 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -2961\) , \( -19684\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-2961{x}-19684$ |
| 14.1-a2 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{14} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$15.75750843$ |
1.101763183 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 5369\) , \( -69762\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+5369{x}-69762$ |
| 14.1-a3 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7^{18} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$5.252502811$ |
1.101763183 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 5614\) , \( -82110\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+5614{x}-82110$ |
| 14.1-a4 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{24} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$2.626251405$ |
1.101763183 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 3654\) , \( -21742\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+3654{x}-21742$ |
| 14.1-a5 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7^{16} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$7.878754216$ |
1.101763183 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 4879\) , \( -45066\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+4879{x}-45066$ |
| 14.1-a6 |
14.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{16} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$0.875417135$ |
1.101763183 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -128401\) , \( -13366500\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-128401{x}-13366500$ |
| 14.1-b1 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \) |
$36.19085529$ |
$1.750834270$ |
5.684544713 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -171\) , \( -874\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-171{x}-874$ |
| 14.1-b2 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \) |
$36.19085529$ |
$15.75750843$ |
5.684544713 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}$ |
| 14.1-b3 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$36.19085529$ |
$5.252502811$ |
5.684544713 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4\) , \( -6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+4{x}-6$ |
| 14.1-b4 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$36.19085529$ |
$2.626251405$ |
5.684544713 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36\) , \( -70\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-36{x}-70$ |
| 14.1-b5 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \) |
$36.19085529$ |
$7.878754216$ |
5.684544713 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -11\) , \( 12\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-11{x}+12$ |
| 14.1-b6 |
14.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2^{3} \) |
$36.19085529$ |
$0.875417135$ |
5.684544713 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2731\) , \( -55146\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-2731{x}-55146$ |
| 14.1-c1 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{36} \cdot 7^{2} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$1.750834270$ |
10.22635756 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 4893\) , \( -61477\bigr] \) |
${y}^2+a{x}{y}={x}^3+{x}^2+4893{x}-61477$ |
| 14.1-c2 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$15.75750843$ |
10.22635756 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 5063\) , \( -69321\bigr] \) |
${y}^2+a{x}{y}={x}^3+{x}^2+5063{x}-69321$ |
| 14.1-c3 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7^{6} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$5.252502811$ |
10.22635756 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 5068\) , \( -69520\bigr] \) |
${y}^2+a{x}{y}={x}^3+{x}^2+5068{x}-69520$ |
| 14.1-c4 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{12} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$2.626251405$ |
10.22635756 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 5028\) , \( -67816\bigr] \) |
${y}^2+a{x}{y}={x}^3+{x}^2+5028{x}-67816$ |
| 14.1-c5 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$7.878754216$ |
10.22635756 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 5053\) , \( -68923\bigr] \) |
${y}^2+a{x}{y}={x}^3+{x}^2+5053{x}-68923$ |
| 14.1-c6 |
14.1-c |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
|
\( 2^{2} \) |
$1$ |
$0.875417135$ |
10.22635756 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 2333\) , \( 97755\bigr] \) |
${y}^2+a{x}{y}={x}^3+{x}^2+2333{x}+97755$ |
| 14.1-d1 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{36} \cdot 7^{14} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \) |
$165.1156049$ |
$1.750834270$ |
25.93492283 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -8355\) , \( 291341\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-8355{x}+291341$ |
| 14.1-d2 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{4} \cdot 7^{14} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \) |
$18.34617832$ |
$15.75750843$ |
25.93492283 |
\( -\frac{15625}{28} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -25\) , \( -111\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-25{x}-111$ |
| 14.1-d3 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{12} \cdot 7^{18} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$18.34617832$ |
$5.252502811$ |
25.93492283 |
\( \frac{9938375}{21952} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 220\) , \( 2192\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2+220{x}+2192$ |
| 14.1-d4 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{6} \cdot 7^{24} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$18.34617832$ |
$2.626251405$ |
25.93492283 |
\( \frac{4956477625}{941192} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -1740\) , \( 22184\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-1740{x}+22184$ |
| 14.1-d5 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{2} \cdot 7^{16} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \) |
$18.34617832$ |
$7.878754216$ |
25.93492283 |
\( \frac{128787625}{98} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -515\) , \( -4717\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-515{x}-4717$ |
| 14.1-d6 |
14.1-d |
$6$ |
$18$ |
\(\Q(\sqrt{-497}) \) |
$2$ |
$[0, 1]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{16} \) |
$7.70688$ |
$(2,a+1), (7,a)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{3} \) |
$165.1156049$ |
$0.875417135$ |
25.93492283 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -133795\) , \( 18781197\bigr] \) |
${y}^2+{x}{y}={x}^3+{x}^2-133795{x}+18781197$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
