| 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 |
| 37.1-a1 |
37.1-a |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{3} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 3 \) |
$3.749312226$ |
$8.952361846$ |
3.928509481 |
\( -\frac{4155745669800443116975209848815}{50653} a^{3} + \frac{7791857257318091093305793370801}{50653} a^{2} + \frac{14480798337431720961859809989470}{50653} a - \frac{2216444873376625079762636975885}{50653} \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a - 5\) , \( a^{2} - a - 4\) , \( 352 a^{3} + 54 a^{2} - 2455 a - 2488\) , \( -2020 a^{3} - 285 a^{2} + 14088 a + 14085\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{3}-5a-5\right){x}^{2}+\left(352a^{3}+54a^{2}-2455a-2488\right){x}-2020a^{3}-285a^{2}+14088a+14085$ |
| 37.1-a2 |
37.1-a |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 1 \) |
$1.249770742$ |
$80.57125662$ |
3.928509481 |
\( -\frac{1234239553}{37} a^{3} + \frac{20463157332}{37} a^{2} + \frac{26028147039}{37} a - \frac{1869517127}{37} \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a - 5\) , \( a^{2} - a - 4\) , \( 217 a^{3} + 34 a^{2} - 1515 a - 1528\) , \( 3150 a^{3} + 446 a^{2} - 21991 a - 22038\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{3}-5a-5\right){x}^{2}+\left(217a^{3}+34a^{2}-1515a-1528\right){x}+3150a^{3}+446a^{2}-21991a-22038$ |
| 37.1-a3 |
37.1-a |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{2} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$2.499541484$ |
$80.57125662$ |
3.928509481 |
\( \frac{5509092}{1369} a^{3} - \frac{16694553}{1369} a^{2} - \frac{33114084}{1369} a - \frac{4668088}{1369} \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a - 5\) , \( a^{2} - a - 4\) , \( 12 a^{3} + 4 a^{2} - 80 a - 83\) , \( 37 a^{3} + 7 a^{2} - 252 a - 254\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{3}-5a-5\right){x}^{2}+\left(12a^{3}+4a^{2}-80a-83\right){x}+37a^{3}+7a^{2}-252a-254$ |
| 37.1-a4 |
37.1-a |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{6} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$7.498624452$ |
$8.952361846$ |
3.928509481 |
\( \frac{9397152679229188098566}{2565726409} a^{3} - \frac{17619286193861002212287}{2565726409} a^{2} - \frac{32744610401601517313878}{2565726409} a + \frac{5011921458100573705269}{2565726409} \) |
\( \bigl[a + 1\) , \( a^{3} - 5 a - 5\) , \( a^{2} - a - 4\) , \( -88 a^{3} - 11 a^{2} + 620 a + 622\) , \( -12 a^{3} + 2 a^{2} + 91 a + 84\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{3}-5a-5\right){x}^{2}+\left(-88a^{3}-11a^{2}+620a+622\right){x}-12a^{3}+2a^{2}+91a+84$ |
| 37.1-b1 |
37.1-b |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 1 \) |
$0.617210121$ |
$129.8037880$ |
3.125631113 |
\( -\frac{1234239553}{37} a^{3} + \frac{20463157332}{37} a^{2} + \frac{26028147039}{37} a - \frac{1869517127}{37} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a + 1\) , \( 4 a^{3} - 8 a^{2} - 12 a - 4\) , \( -7 a^{3} + 12 a^{2} + 24 a - 8\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(4a^{3}-8a^{2}-12a-4\right){x}-7a^{3}+12a^{2}+24a-8$ |
| 37.1-b2 |
37.1-b |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{2} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1.234420243$ |
$129.8037880$ |
3.125631113 |
\( \frac{5509092}{1369} a^{3} - \frac{16694553}{1369} a^{2} - \frac{33114084}{1369} a - \frac{4668088}{1369} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a + 1\) , \( -a^{3} + 2 a^{2} + 3 a + 1\) , \( -a^{3} + 2 a^{2} + 4 a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(-a^{3}+2a^{2}+3a+1\right){x}-a^{3}+2a^{2}+4a-1$ |
| 37.1-b3 |
37.1-b |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{6} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.703260729$ |
$129.8037880$ |
3.125631113 |
\( \frac{9397152679229188098566}{2565726409} a^{3} - \frac{17619286193861002212287}{2565726409} a^{2} - \frac{32744610401601517313878}{2565726409} a + \frac{5011921458100573705269}{2565726409} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a + 1\) , \( 39 a^{3} - 73 a^{2} - 137 a + 26\) , \( -153 a^{3} + 289 a^{2} + 529 a - 83\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(39a^{3}-73a^{2}-137a+26\right){x}-153a^{3}+289a^{2}+529a-83$ |
| 37.1-b4 |
37.1-b |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{3} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 3 \) |
$1.851630364$ |
$129.8037880$ |
3.125631113 |
\( -\frac{4155745669800443116975209848815}{50653} a^{3} + \frac{7791857257318091093305793370801}{50653} a^{2} + \frac{14480798337431720961859809989470}{50653} a - \frac{2216444873376625079762636975885}{50653} \) |
\( \bigl[a^{3} - a^{2} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 3\) , \( a + 1\) , \( 619 a^{3} - 1168 a^{2} - 2132 a + 316\) , \( -13831 a^{3} + 25966 a^{2} + 48081 a - 7351\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-3\right){x}^{2}+\left(619a^{3}-1168a^{2}-2132a+316\right){x}-13831a^{3}+25966a^{2}+48081a-7351$ |
| 37.1-c1 |
37.1-c |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{3} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 3 \) |
$0.221644447$ |
$147.5700806$ |
3.828192253 |
\( -\frac{4155745669800443116975209848815}{50653} a^{3} + \frac{7791857257318091093305793370801}{50653} a^{2} + \frac{14480798337431720961859809989470}{50653} a - \frac{2216444873376625079762636975885}{50653} \) |
\( \bigl[a^{2} - 4\) , \( -a^{3} + 6 a + 4\) , \( a^{2} - 4\) , \( 350 a^{3} + 52 a^{2} - 2448 a - 2483\) , \( 2424 a^{3} + 344 a^{2} - 16921 a - 16927\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{3}+6a+4\right){x}^{2}+\left(350a^{3}+52a^{2}-2448a-2483\right){x}+2424a^{3}+344a^{2}-16921a-16927$ |
| 37.1-c2 |
37.1-c |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 1 \) |
$0.073881482$ |
$1328.130725$ |
3.828192253 |
\( -\frac{1234239553}{37} a^{3} + \frac{20463157332}{37} a^{2} + \frac{26028147039}{37} a - \frac{1869517127}{37} \) |
\( \bigl[a^{2} - 4\) , \( -a^{3} + 6 a + 4\) , \( a^{2} - 4\) , \( 215 a^{3} + 32 a^{2} - 1508 a - 1523\) , \( -2901 a^{3} - 412 a^{2} + 20248 a + 20291\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{3}+6a+4\right){x}^{2}+\left(215a^{3}+32a^{2}-1508a-1523\right){x}-2901a^{3}-412a^{2}+20248a+20291$ |
| 37.1-c3 |
37.1-c |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{2} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.147762964$ |
$1328.130725$ |
3.828192253 |
\( \frac{5509092}{1369} a^{3} - \frac{16694553}{1369} a^{2} - \frac{33114084}{1369} a - \frac{4668088}{1369} \) |
\( \bigl[a^{2} - 4\) , \( -a^{3} + 6 a + 4\) , \( a^{2} - 4\) , \( 10 a^{3} + 2 a^{2} - 73 a - 78\) , \( -23 a^{3} - 3 a^{2} + 159 a + 157\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{3}+6a+4\right){x}^{2}+\left(10a^{3}+2a^{2}-73a-78\right){x}-23a^{3}-3a^{2}+159a+157$ |
| 37.1-c4 |
37.1-c |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{6} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.443288894$ |
$147.5700806$ |
3.828192253 |
\( \frac{9397152679229188098566}{2565726409} a^{3} - \frac{17619286193861002212287}{2565726409} a^{2} - \frac{32744610401601517313878}{2565726409} a + \frac{5011921458100573705269}{2565726409} \) |
\( \bigl[a^{2} - 4\) , \( -a^{3} + 6 a + 4\) , \( a^{2} - 4\) , \( -90 a^{3} - 13 a^{2} + 627 a + 627\) , \( -89 a^{3} - 13 a^{2} + 621 a + 624\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{3}+6a+4\right){x}^{2}+\left(-90a^{3}-13a^{2}+627a+627\right){x}-89a^{3}-13a^{2}+621a+624$ |
| 37.1-d1 |
37.1-d |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 1 \) |
$0.832260033$ |
$824.3916693$ |
2.974181307 |
\( -\frac{1234239553}{37} a^{3} + \frac{20463157332}{37} a^{2} + \frac{26028147039}{37} a - \frac{1869517127}{37} \) |
\( \bigl[a^{3} - 5 a - 4\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( 1\) , \( 3 a^{3} - 4 a^{2}\) , \( 42 a^{3} - 71 a^{2} - 151 a + 23\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(3a^{3}-4a^{2}\right){x}+42a^{3}-71a^{2}-151a+23$ |
| 37.1-d2 |
37.1-d |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{2} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1.664520066$ |
$824.3916693$ |
2.974181307 |
\( \frac{5509092}{1369} a^{3} - \frac{16694553}{1369} a^{2} - \frac{33114084}{1369} a - \frac{4668088}{1369} \) |
\( \bigl[a^{3} - 5 a - 4\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( 1\) , \( -2 a^{3} + 6 a^{2} + 15 a + 5\) , \( a^{3} + 4 a^{2} + 4 a + 1\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(-2a^{3}+6a^{2}+15a+5\right){x}+a^{3}+4a^{2}+4a+1$ |
| 37.1-d3 |
37.1-d |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{6} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$4.993560199$ |
$10.17767493$ |
2.974181307 |
\( \frac{9397152679229188098566}{2565726409} a^{3} - \frac{17619286193861002212287}{2565726409} a^{2} - \frac{32744610401601517313878}{2565726409} a + \frac{5011921458100573705269}{2565726409} \) |
\( \bigl[a^{3} - 5 a - 4\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( 1\) , \( 38 a^{3} - 69 a^{2} - 125 a + 30\) , \( 318 a^{3} - 593 a^{2} - 1091 a + 173\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(38a^{3}-69a^{2}-125a+30\right){x}+318a^{3}-593a^{2}-1091a+173$ |
| 37.1-d4 |
37.1-d |
$4$ |
$6$ |
4.4.10512.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37^{3} \) |
$14.38820$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 3 \) |
$2.496780099$ |
$10.17767493$ |
2.974181307 |
\( -\frac{4155745669800443116975209848815}{50653} a^{3} + \frac{7791857257318091093305793370801}{50653} a^{2} + \frac{14480798337431720961859809989470}{50653} a - \frac{2216444873376625079762636975885}{50653} \) |
\( \bigl[a^{3} - 5 a - 4\) , \( -a^{3} + 2 a^{2} + 3 a - 2\) , \( 1\) , \( 618 a^{3} - 1164 a^{2} - 2120 a + 320\) , \( 16546 a^{3} - 31040 a^{2} - 57578 a + 8796\bigr] \) |
${y}^2+\left(a^{3}-5a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-2\right){x}^{2}+\left(618a^{3}-1164a^{2}-2120a+320\right){x}+16546a^{3}-31040a^{2}-57578a+8796$ |
*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.
