| 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 |
| 8.1-a1 |
8.1-a |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{9} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B.1.1, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$182.6574446$ |
0.670574649 |
\( \frac{4847}{32} a^{2} - \frac{17157}{32} a + \frac{14147}{32} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} - a + 4\) , \( a\) , \( -3 a^{2} + a + 9\) , \( -3 a^{2} + 3 a + 6\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-3a^{2}+a+9\right){x}-3a^{2}+3a+6$ |
| 8.1-a2 |
8.1-a |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{27} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B.1.2, 5B.4.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.765090543$ |
0.670574649 |
\( \frac{5149908848399}{32768} a^{2} - \frac{9582969548701}{32768} a - \frac{2767591890197}{32768} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} - a + 4\) , \( a\) , \( -78 a^{2} + 141 a + 49\) , \( -725 a^{2} + 1346 a + 393\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-78a^{2}+141a+49\right){x}-725a^{2}+1346a+393$ |
| 8.1-a3 |
8.1-a |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{21} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B.1.1, 5B.4.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$36.53148893$ |
0.670574649 |
\( \frac{410180291815113}{1024} a^{2} - \frac{381468430679405}{512} a - \frac{221127400043935}{1024} \) |
\( \bigl[a^{2} - 2\) , \( -a^{2} - a + 4\) , \( a\) , \( -403 a^{2} + 746 a + 214\) , \( 7473 a^{2} - 13910 a - 4017\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-403a^{2}+746a+214\right){x}+7473a^{2}-13910a-4017$ |
| 8.1-a4 |
8.1-a |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{12} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B.1.1, 5B.4.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$36.53148893$ |
0.670574649 |
\( -\frac{304458198456664063}{2} a^{2} + \frac{1237813476398146963}{32} a + \frac{19170794207295639649}{32} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{2} - 2\) , \( 1227 a^{2} - 425 a - 5034\) , \( -35239 a^{2} + 7997 a + 136900\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(1227a^{2}-425a-5034\right){x}-35239a^{2}+7997a+136900$ |
| 8.1-a5 |
8.1-a |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{36} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B.1.2, 5B.4.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.765090543$ |
0.670574649 |
\( -\frac{550001475734697}{1073741824} a^{2} + \frac{154648619367179}{1073741824} a + \frac{2158976306071891}{1073741824} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{2} - 2\) , \( 52 a^{2} - 30 a - 229\) , \( 275 a^{2} - 119 a - 1172\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(52a^{2}-30a-229\right){x}+275a^{2}-119a-1172$ |
| 8.1-a6 |
8.1-a |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{12} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B.1.1, 5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$182.6574446$ |
0.670574649 |
\( \frac{13210743}{1024} a^{2} + \frac{29758379}{1024} a + \frac{11183859}{1024} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{2} - 2\) , \( 2 a^{2} - 5 a - 9\) , \( -2 a^{2} + a + 11\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(2a^{2}-5a-9\right){x}-2a^{2}+a+11$ |
| 8.1-a7 |
8.1-a |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{63} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B.1.2, 5B.4.2 |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$1$ |
$1.353018108$ |
0.670574649 |
\( -\frac{9591905236373162715}{1073741824} a^{2} - \frac{20286002608069784023}{1073741824} a - \frac{2267690473188435537}{536870912} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 433 a^{2} - 66 a - 1835\) , \( -18127 a^{2} + 4359 a + 71774\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(433a^{2}-66a-1835\right){x}-18127a^{2}+4359a+71774$ |
| 8.1-a8 |
8.1-a |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{36} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B.1.2, 5B.4.2 |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$1$ |
$1.353018108$ |
0.670574649 |
\( \frac{24628729701449988584212043}{32768} a^{2} + \frac{52087486182589166202672597}{32768} a + \frac{2911324634581045729032995}{8192} \) |
\( \bigl[1\) , \( a^{2} - 4\) , \( a + 1\) , \( 782 a^{2} - 1290 a - 896\) , \( 79104 a^{2} - 149643 a - 37909\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(782a^{2}-1290a-896\right){x}+79104a^{2}-149643a-37909$ |
| 8.1-b1 |
8.1-b |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{36} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B, 5B.1.2 |
$25$ |
\( 2 \cdot 3 \) |
$1$ |
$0.411398833$ |
1.019475005 |
\( \frac{24628729701449988584212043}{32768} a^{2} + \frac{52087486182589166202672597}{32768} a + \frac{2911324634581045729032995}{8192} \) |
\( \bigl[a^{2} - 2\) , \( 0\) , \( a + 1\) , \( 2232 a^{2} - 4377 a - 1289\) , \( -519466 a^{2} + 964390 a + 278508\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(2232a^{2}-4377a-1289\right){x}-519466a^{2}+964390a+278508$ |
| 8.1-b2 |
8.1-b |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{21} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B, 5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$1.234196500$ |
1.019475005 |
\( \frac{410180291815113}{1024} a^{2} - \frac{381468430679405}{512} a - \frac{221127400043935}{1024} \) |
\( \bigl[1\) , \( -a^{2} - a + 3\) , \( a\) , \( -76 a^{2} + 206 a - 95\) , \( -1041 a^{2} + 2329 a - 271\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-76a^{2}+206a-95\right){x}-1041a^{2}+2329a-271$ |
| 8.1-b3 |
8.1-b |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{63} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B, 5B.1.2 |
$25$ |
\( 2 \cdot 3 \) |
$1$ |
$0.411398833$ |
1.019475005 |
\( -\frac{9591905236373162715}{1073741824} a^{2} - \frac{20286002608069784023}{1073741824} a - \frac{2267690473188435537}{536870912} \) |
\( \bigl[1\) , \( -a^{2} - a + 3\) , \( a\) , \( -106 a^{2} + 176 a - 60\) , \( -971 a^{2} + 2082 a - 692\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-106a^{2}+176a-60\right){x}-971a^{2}+2082a-692$ |
| 8.1-b4 |
8.1-b |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{9} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B, 5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$154.2745625$ |
1.019475005 |
\( \frac{4847}{32} a^{2} - \frac{17157}{32} a + \frac{14147}{32} \) |
\( \bigl[1\) , \( -a^{2} - a + 3\) , \( a\) , \( -a^{2} + a + 5\) , \( -a^{2} + 2\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-a^{2}+a+5\right){x}-a^{2}+2$ |
| 8.1-b5 |
8.1-b |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{27} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B, 5B.1.1 |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$1$ |
$51.42485419$ |
1.019475005 |
\( \frac{5149908848399}{32768} a^{2} - \frac{9582969548701}{32768} a - \frac{2767591890197}{32768} \) |
\( \bigl[1\) , \( -a^{2} - a + 3\) , \( a\) , \( -21 a^{2} + 41 a + 10\) , \( 87 a^{2} - 165 a - 43\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-21a^{2}+41a+10\right){x}+87a^{2}-165a-43$ |
| 8.1-b6 |
8.1-b |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{12} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B, 5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$1.234196500$ |
1.019475005 |
\( -\frac{304458198456664063}{2} a^{2} + \frac{1237813476398146963}{32} a + \frac{19170794207295639649}{32} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a^{2} + a - 3\) , \( -131 a^{2} - 464 a - 415\) , \( -2913 a^{2} - 7754 a - 4350\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-131a^{2}-464a-415\right){x}-2913a^{2}-7754a-4350$ |
| 8.1-b7 |
8.1-b |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{36} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B, 5B.1.1 |
$1$ |
\( 2 \cdot 3 \cdot 5 \) |
$1$ |
$51.42485419$ |
1.019475005 |
\( -\frac{550001475734697}{1073741824} a^{2} + \frac{154648619367179}{1073741824} a + \frac{2158976306071891}{1073741824} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a^{2} + a - 3\) , \( -26 a^{2} - 59 a - 20\) , \( -163 a^{2} - 344 a - 76\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-26a^{2}-59a-20\right){x}-163a^{2}-344a-76$ |
| 8.1-b8 |
8.1-b |
$8$ |
$30$ |
3.3.229.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{12} \) |
$1.91237$ |
$(a+1), (a^2-a-3)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3, 5$ |
2B, 3B, 5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$154.2745625$ |
1.019475005 |
\( \frac{13210743}{1024} a^{2} + \frac{29758379}{1024} a + \frac{11183859}{1024} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( a^{2} + a - 3\) , \( -6 a^{2} - 9 a + 5\) , \( 10 a^{2} + 21 a + 4\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-6a^{2}-9a+5\right){x}+10a^{2}+21a+4$ |
*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.
