| 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 |
| 39.1-a1 |
39.1-a |
$4$ |
$4$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{32} \cdot 13 \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$67.81533581$ |
1.967522449 |
\( \frac{39345837316936361}{85293} a^{3} + \frac{19496166946180265}{85293} a^{2} - \frac{5337283778507008}{6561} a - \frac{5255555751196786}{28431} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{2} + 2\) , \( a^{3} - a^{2} - 2 a + 1\) , \( -a^{3} + 4 a^{2} + 19 a - 89\) , \( 22 a^{3} - 125 a^{2} + 144 a + 85\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-a^{3}+4a^{2}+19a-89\right){x}+22a^{3}-125a^{2}+144a+85$ |
| 39.1-a2 |
39.1-a |
$4$ |
$4$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{16} \cdot 13^{2} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$67.81533581$ |
1.967522449 |
\( -\frac{5413523923220}{4563} a^{3} + \frac{56779324706711}{13689} a^{2} - \frac{927897100844}{351} a - \frac{10861161495304}{13689} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{2} + 2\) , \( a^{3} - a^{2} - 2 a + 1\) , \( -a^{3} - 6 a^{2} + 34 a - 44\) , \( 4 a^{3} - 46 a^{2} + 147 a - 149\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-a^{3}-6a^{2}+34a-44\right){x}+4a^{3}-46a^{2}+147a-149$ |
| 39.1-a3 |
39.1-a |
$4$ |
$4$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{8} \cdot 13^{4} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$67.81533581$ |
1.967522449 |
\( \frac{2332444130}{257049} a^{3} - \frac{8270916053}{257049} a^{2} + \frac{413334676}{19773} a + \frac{1603089799}{257049} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{2} + 2\) , \( a^{3} - a^{2} - 2 a + 1\) , \( -a^{3} - a^{2} + 4 a + 1\) , \( -a^{3} + 6 a - 5\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-a^{3}-a^{2}+4a+1\right){x}-a^{3}+6a-5$ |
| 39.1-a4 |
39.1-a |
$4$ |
$4$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{8} \cdot 13 \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$4.238458488$ |
1.967522449 |
\( -\frac{2277542037635157183569}{117} a^{3} + \frac{7961165631081581255087}{117} a^{2} - \frac{390258311116918949632}{9} a - \frac{1522922350984212886954}{117} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 2 a + 1\) , \( -1385 a^{3} - 686 a^{2} + 2441 a + 527\) , \( 40164 a^{3} + 19914 a^{2} - 70813 a - 16170\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(-1385a^{3}-686a^{2}+2441a+527\right){x}+40164a^{3}+19914a^{2}-70813a-16170$ |
| 39.1-b1 |
39.1-b |
$4$ |
$14$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{7} \cdot 13^{2} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.6.1 |
$1$ |
\( 2 \) |
$1$ |
$112.0379970$ |
0.812637996 |
\( -\frac{5051534848}{1521} a^{3} + \frac{17683689664}{1521} a^{2} - \frac{869445632}{117} a - \frac{3355268992}{1521} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 3 a\) , \( a^{3} - 3 a - 1\) , \( -2 a^{3} - 3 a^{2} + 6 a + 1\) , \( -a^{3} - 4 a^{2} + 4 a + 1\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-2a^{3}-3a^{2}+6a+1\right){x}-a^{3}-4a^{2}+4a+1$ |
| 39.1-b2 |
39.1-b |
$4$ |
$14$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13^{14} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.6.3 |
$1$ |
\( 2 \) |
$1$ |
$112.0379970$ |
0.812637996 |
\( \frac{2547185536653029069094939392}{11812129157097867} a^{3} - \frac{1701087144546817548515403072}{3937376385699289} a^{2} - \frac{160194694469438750717543424}{302875106592253} a + \frac{10590159144336371732057734784}{11812129157097867} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + 3 a\) , \( a^{3} - 3 a - 1\) , \( -797 a^{3} - 418 a^{2} + 1491 a + 191\) , \( 34097 a^{3} + 17116 a^{2} - 60757 a - 13092\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(-797a^{3}-418a^{2}+1491a+191\right){x}+34097a^{3}+17116a^{2}-60757a-13092$ |
| 39.1-b3 |
39.1-b |
$4$ |
$14$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13^{7} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.6.3 |
$1$ |
\( 2 \) |
$1$ |
$112.0379970$ |
0.812637996 |
\( \frac{767268311968017524755828572928}{188245551} a^{3} + \frac{380187323252878675163452774720}{188245551} a^{2} - \frac{104080350752632893030271750528}{14480427} a - \frac{307459810806554562884410540096}{188245551} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - a^{2} - 2 a + 1\) , \( -1364 a^{3} - 115 a^{2} + 2627 a - 566\) , \( 57878 a^{3} + 21925 a^{2} - 104397 a - 10292\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(-1364a^{3}-115a^{2}+2627a-566\right){x}+57878a^{3}+21925a^{2}-104397a-10292$ |
| 39.1-b4 |
39.1-b |
$4$ |
$14$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{14} \cdot 13 \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.6.1 |
$1$ |
\( 2 \) |
$1$ |
$112.0379970$ |
0.812637996 |
\( -\frac{15263744}{1053} a^{3} + \frac{10906816}{1053} a^{2} + \frac{4893824}{81} a + \frac{13306688}{1053} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -1\) , \( a + 1\) , \( 2 a^{3} - 2 a^{2} - 9 a - 2\) , \( 3 a^{3} - 2 a^{2} - 13 a - 5\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(2a^{3}-2a^{2}-9a-2\right){x}+3a^{3}-2a^{2}-13a-5$ |
| 39.1-c1 |
39.1-c |
$2$ |
$2$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13 \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$224.6931443$ |
1.629752328 |
\( -\frac{14454196736}{39} a^{3} + \frac{11278831936}{39} a^{2} + \frac{4393788032}{3} a + \frac{11850812096}{39} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{2} + 1\) , \( a^{3} - 3 a\) , \( -2 a^{3} - a^{2} + 3 a + 2\) , \( -2 a^{3} - a^{2} + 4 a + 1\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-2a^{3}-a^{2}+3a+2\right){x}-2a^{3}-a^{2}+4a+1$ |
| 39.1-c2 |
39.1-c |
$2$ |
$2$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13^{2} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$224.6931443$ |
1.629752328 |
\( \frac{3245698304}{507} a^{3} + \frac{534167488}{169} a^{2} - \frac{147011840}{13} a - \frac{1301490304}{507} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a - 1\) , \( a\) , \( -3 a^{3} + 7 a^{2} + 7 a - 13\) , \( -2 a^{3} + 4 a^{2} + 5 a - 9\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3a^{3}+7a^{2}+7a-13\right){x}-2a^{3}+4a^{2}+5a-9$ |
| 39.1-d1 |
39.1-d |
$2$ |
$2$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13 \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.047761652$ |
$1167.620777$ |
1.617979207 |
\( -\frac{14454196736}{39} a^{3} + \frac{11278831936}{39} a^{2} + \frac{4393788032}{3} a + \frac{11850812096}{39} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} + a^{2} - 3 a + 1\) , \( a^{3} - a\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(a^{3}+a^{2}-3a+1\right){x}+a^{3}-a$ |
| 39.1-d2 |
39.1-d |
$2$ |
$2$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13^{2} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.023880826$ |
$2335.241554$ |
1.617979207 |
\( \frac{3245698304}{507} a^{3} + \frac{534167488}{169} a^{2} - \frac{147011840}{13} a - \frac{1301490304}{507} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} - 1\) , \( -4 a^{3} + 6 a^{2} + 10 a - 11\) , \( 2 a^{3} - 6 a^{2} - 5 a + 12\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-4a^{3}+6a^{2}+10a-11\right){x}+2a^{3}-6a^{2}-5a+12$ |
| 39.1-e1 |
39.1-e |
$4$ |
$14$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{2} \cdot 13^{7} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.1.3 |
$49$ |
\( 2 \) |
$5.198535081$ |
$0.312872496$ |
2.312257658 |
\( \frac{767268311968017524755828572928}{188245551} a^{3} + \frac{380187323252878675163452774720}{188245551} a^{2} - \frac{104080350752632893030271750528}{14480427} a - \frac{307459810806554562884410540096}{188245551} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( a + 1\) , \( -1366 a^{3} - 112 a^{2} + 2635 a - 571\) , \( -59243 a^{3} - 22039 a^{2} + 107028 a + 9722\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-1\right){x}^{2}+\left(-1366a^{3}-112a^{2}+2635a-571\right){x}-59243a^{3}-22039a^{2}+107028a+9722$ |
| 39.1-e2 |
39.1-e |
$4$ |
$14$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3^{7} \cdot 13^{2} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/14\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.1.1 |
$1$ |
\( 2 \cdot 7 \) |
$0.371323934$ |
$1502.413729$ |
2.312257658 |
\( -\frac{5051534848}{1521} a^{3} + \frac{17683689664}{1521} a^{2} - \frac{869445632}{117} a - \frac{3355268992}{1521} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - a^{2} - 2 a\) , \( a^{2} - 1\) , \( -2 a^{3} - 3 a^{2} + 7 a + 1\) , \( 3 a^{2} - 3 a - 1\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a\right){x}^{2}+\left(-2a^{3}-3a^{2}+7a+1\right){x}+3a^{2}-3a-1$ |
| 39.1-e3 |
39.1-e |
$4$ |
$14$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( - 3 \cdot 13^{14} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.1.3 |
$49$ |
\( 2 \) |
$2.599267540$ |
$0.625744993$ |
2.312257658 |
\( \frac{2547185536653029069094939392}{11812129157097867} a^{3} - \frac{1701087144546817548515403072}{3937376385699289} a^{2} - \frac{160194694469438750717543424}{302875106592253} a + \frac{10590159144336371732057734784}{11812129157097867} \) |
\( \bigl[a^{2} - a - 1\) , \( a^{3} - a^{2} - 2 a\) , \( a^{2} - 1\) , \( -797 a^{3} - 418 a^{2} + 1492 a + 191\) , \( -34098 a^{3} - 17117 a^{2} + 60758 a + 13092\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a\right){x}^{2}+\left(-797a^{3}-418a^{2}+1492a+191\right){x}-34098a^{3}-17117a^{2}+60758a+13092$ |
| 39.1-e4 |
39.1-e |
$4$ |
$14$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{14} \cdot 13 \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/14\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 7$ |
2B, 7B.1.1 |
$1$ |
\( 2 \cdot 7 \) |
$0.742647868$ |
$751.2068645$ |
2.312257658 |
\( -\frac{15263744}{1053} a^{3} + \frac{10906816}{1053} a^{2} + \frac{4893824}{81} a + \frac{13306688}{1053} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 2 a - 2\) , \( a^{3} - 3 a\) , \( 2 a^{3} - 3 a^{2} - 10 a - 2\) , \( -4 a^{3} + a^{2} + 14 a + 5\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(2a^{3}-3a^{2}-10a-2\right){x}-4a^{3}+a^{2}+14a+5$ |
| 39.1-f1 |
39.1-f |
$4$ |
$4$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{8} \cdot 13 \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.143231163$ |
$272.5456136$ |
2.259982308 |
\( -\frac{2277542037635157183569}{117} a^{3} + \frac{7961165631081581255087}{117} a^{2} - \frac{390258311116918949632}{9} a - \frac{1522922350984212886954}{117} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} - 2\) , \( -1385 a^{3} - 686 a^{2} + 2441 a + 526\) , \( -40165 a^{3} - 19914 a^{2} + 70815 a + 16169\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-1385a^{3}-686a^{2}+2441a+526\right){x}-40165a^{3}-19914a^{2}+70815a+16169$ |
| 39.1-f2 |
39.1-f |
$4$ |
$4$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{32} \cdot 13 \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1.143231163$ |
$17.03410085$ |
2.259982308 |
\( \frac{39345837316936361}{85293} a^{3} + \frac{19496166946180265}{85293} a^{2} - \frac{5337283778507008}{6561} a - \frac{5255555751196786}{28431} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{3} - 3 a - 1\) , \( 3 a^{2} + 16 a - 90\) , \( -23 a^{3} + 128 a^{2} - 125 a - 175\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}-a{x}^{2}+\left(3a^{2}+16a-90\right){x}-23a^{3}+128a^{2}-125a-175$ |
| 39.1-f3 |
39.1-f |
$4$ |
$4$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{16} \cdot 13^{2} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.571615581$ |
$272.5456136$ |
2.259982308 |
\( -\frac{5413523923220}{4563} a^{3} + \frac{56779324706711}{13689} a^{2} - \frac{927897100844}{351} a - \frac{10861161495304}{13689} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{3} - 3 a - 1\) , \( -7 a^{2} + 31 a - 45\) , \( -5 a^{3} + 39 a^{2} - 113 a + 104\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}-a{x}^{2}+\left(-7a^{2}+31a-45\right){x}-5a^{3}+39a^{2}-113a+104$ |
| 39.1-f4 |
39.1-f |
$4$ |
$4$ |
4.4.4752.1 |
$4$ |
$[4, 0]$ |
39.1 |
\( 3 \cdot 13 \) |
\( 3^{8} \cdot 13^{4} \) |
$9.73778$ |
$(a^3-a^2-4a+1), (-a^3+a^2+2a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.285807790$ |
$272.5456136$ |
2.259982308 |
\( \frac{2332444130}{257049} a^{3} - \frac{8270916053}{257049} a^{2} + \frac{413334676}{19773} a + \frac{1603089799}{257049} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{3} - 3 a - 1\) , \( -2 a^{2} + a\) , \( -2 a^{2} - 2 a + 5\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}-a{x}^{2}+\left(-2a^{2}+a\right){x}-2a^{2}-2a+5$ |
*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.
