| 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 |
| 14.3-a2 |
14.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
14.3 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{9} \) |
$0.96239$ |
$(2,a+1), (7,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$1.122170831$ |
0.806191324 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 29\) , \( 24 a - 66\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+29{x}+24a-66$ |
| 350.11-a2 |
350.11-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
350.11 |
\( 2 \cdot 5^{2} \cdot 7 \) |
\( 2^{30} \cdot 5^{6} \cdot 7^{9} \) |
$2.15197$ |
$(2,a+1), (5,a+3), (7,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$0.882828228$ |
$0.501850052$ |
1.273178572 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 210 a + 38\) , \( 2459 a + 3780\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+a{x}^2+\left(210a+38\right){x}+2459a+3780$ |
| 350.7-c2 |
350.7-c |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
350.7 |
\( 2 \cdot 5^{2} \cdot 7 \) |
\( 2^{30} \cdot 5^{6} \cdot 7^{9} \) |
$2.15197$ |
$(2,a+1), (5,a+1), (7,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \cdot 3^{4} \) |
$0.075602030$ |
$0.501850052$ |
4.415720667 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -213 a + 232\) , \( 195 a - 8088\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+a{x}^2+\left(-213a+232\right){x}+195a-8088$ |
| 448.13-b2 |
448.13-b |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
448.13 |
\( 2^{6} \cdot 7 \) |
\( 2^{36} \cdot 7^{9} \) |
$2.28896$ |
$(2,a+1), (7,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$3.801117233$ |
$0.396747302$ |
2.166877634 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -29 a - 215\) , \( -715 a - 774\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-29a-215\right){x}-715a-774$ |
| 784.13-b2 |
784.13-b |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
784.13 |
\( 2^{4} \cdot 7^{2} \) |
\( 2^{30} \cdot 7^{15} \) |
$2.63268$ |
$(2,a+1), (7,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$6.717273526$ |
$0.212070353$ |
4.093663392 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( -272 a + 497\) , \( 1688 a + 13511\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(-272a+497\right){x}+1688a+13511$ |
| 896.3-c2 |
896.3-c |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
896.3 |
\( 2^{7} \cdot 7 \) |
\( 2^{36} \cdot 7^{9} \) |
$2.72205$ |
$(2,a), (2,a+1), (7,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$3.370318600$ |
$0.793494604$ |
3.842590242 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 29 a - 237\) , \( -199 a + 2349\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a-1\right){x}^2+\left(29a-237\right){x}-199a+2349$ |
| 1134.3-a2 |
1134.3-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1134.3 |
\( 2 \cdot 3^{4} \cdot 7 \) |
\( 2^{18} \cdot 3^{12} \cdot 7^{9} \) |
$2.88717$ |
$(2,a+1), (7,a+2), (3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{4} \) |
$0.889555971$ |
$0.374056943$ |
4.302913844 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -a + 277\) , \( -917 a + 1237\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-a+277\right){x}-917a+1237$ |
| 1568.4-d2 |
1568.4-d |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1568.4 |
\( 2^{5} \cdot 7^{2} \) |
\( 2^{42} \cdot 7^{15} \) |
$3.13080$ |
$(2,a), (2,a+1), (7,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$0.212070353$ |
5.484810226 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -1167 a - 296\) , \( -26145 a + 96824\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-a{x}^2+\left(-1167a-296\right){x}-26145a+96824$ |
| 1792.9-b2 |
1792.9-b |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
1792.9 |
\( 2^{8} \cdot 7 \) |
\( 2^{42} \cdot 7^{9} \) |
$3.23708$ |
$(2,a), (2,a+1), (7,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$15.36121444$ |
$0.280542707$ |
6.192038910 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -3 a + 483\) , \( -2015 a + 3739\bigr] \) |
${y}^2={x}^3+{x}^2+\left(-3a+483\right){x}-2015a+3739$ |
| 3136.19-d2 |
3136.19-d |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
3136.19 |
\( 2^{6} \cdot 7^{2} \) |
\( 2^{48} \cdot 7^{15} \) |
$3.72317$ |
$(2,a+1), (7,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{4} \) |
$1.673970490$ |
$0.149956385$ |
2.885438934 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( 1685 a - 5682\) , \( 76008 a - 280188\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3-a{x}^2+\left(1685a-5682\right){x}+76008a-280188$ |
| 4802.8-a2 |
4802.8-a |
$4$ |
$27$ |
\(\Q(\sqrt{-31}) \) |
$2$ |
$[0, 1]$ |
4802.8 |
\( 2 \cdot 7^{4} \) |
\( 2^{18} \cdot 7^{21} \) |
$4.14166$ |
$(2,a+1), (7,a+2), (7,a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2^{3} \) |
$2.038730963$ |
$0.160310118$ |
1.878408247 |
\( \frac{3429643149944533}{10578455953408} a - \frac{483319425333757}{1322306994176} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -9 a + 1482\) , \( -11156 a + 18175\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-9a+1482\right){x}-11156a+18175$ |
*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.
