| 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 |
| 1156.1-a1 |
1156.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{6} \cdot 17^{2} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1.882925266$ |
$2.954600273$ |
3.085958890 |
\( \frac{1371564592745}{68} a - \frac{6316809440783}{136} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 36 a - 88\) , \( 181 a - 417\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(36a-88\right){x}+181a-417$ |
| 1156.1-a2 |
1156.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{2} \cdot 17^{6} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.627641755$ |
$2.954600273$ |
3.085958890 |
\( \frac{913589569}{9826} a + \frac{578023451}{4913} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -12 a - 19\) , \( -37 a - 55\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-12a-19\right){x}-37a-55$ |
| 1156.1-b1 |
1156.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{2} \cdot 17^{2} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.059716613$ |
$29.26266179$ |
0.969320326 |
\( \frac{34373}{34} a + \frac{48885}{34} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -3 a - 4\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-3a-4\right){x}$ |
| 1156.1-c1 |
1156.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{14} \cdot 17^{8} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$3.337638858$ |
0.925694464 |
\( -\frac{49709529748121}{26261675072} a + \frac{274513628114411}{52523350144} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 195 a - 445\) , \( -1721 a + 3966\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(195a-445\right){x}-1721a+3966$ |
| 1156.1-d1 |
1156.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{2} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.846879378$ |
$20.21098874$ |
1.582399683 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-3{x}+1$ |
| 1156.1-d2 |
1156.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{2} \cdot 17^{12} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1.270319068$ |
$2.245665415$ |
1.582399683 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \) |
${y}^2+{x}{y}={x}^{3}-113{x}-329$ |
| 1156.1-d3 |
1156.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{4} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.423439689$ |
$20.21098874$ |
1.582399683 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \) |
${y}^2+{x}{y}={x}^{3}-43{x}+105$ |
| 1156.1-d4 |
1156.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( 2^{4} \cdot 17^{6} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$2.540638136$ |
$2.245665415$ |
1.582399683 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \) |
${y}^2+{x}{y}={x}^{3}-103{x}-411$ |
| 1156.1-e1 |
1156.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{6} \cdot 17^{2} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1.882925266$ |
$2.954600273$ |
3.085958890 |
\( -\frac{1371564592745}{68} a - \frac{3573680255293}{136} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -36 a - 55\) , \( -146 a - 184\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-36a-55\right){x}-146a-184$ |
| 1156.1-e2 |
1156.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{2} \cdot 17^{6} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.627641755$ |
$2.954600273$ |
3.085958890 |
\( -\frac{913589569}{9826} a + \frac{2069636471}{9826} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 11 a - 31\) , \( 36 a - 92\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(11a-31\right){x}+36a-92$ |
| 1156.1-f1 |
1156.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{6} \cdot 17^{6} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \cdot 5 \) |
$0.019399282$ |
$11.63336164$ |
1.877761721 |
\( \frac{487724602833}{11358856} a - \frac{550943316729}{5679428} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -49 a - 66\) , \( 82 a + 108\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-49a-66\right){x}+82a+108$ |
| 1156.1-g1 |
1156.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{14} \cdot 17^{8} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$3.337638858$ |
0.925694464 |
\( \frac{49709529748121}{26261675072} a + \frac{175094568618169}{52523350144} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -196 a - 249\) , \( 1720 a + 2246\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-196a-249\right){x}+1720a+2246$ |
| 1156.1-h1 |
1156.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{2} \cdot 17^{2} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.059716613$ |
$29.26266179$ |
0.969320326 |
\( -\frac{34373}{34} a + \frac{41629}{17} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( a - 6\) , \( -a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(a-6\right){x}-a+1$ |
| 1156.1-i1 |
1156.1-i |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1156.1 |
\( 2^{2} \cdot 17^{2} \) |
\( - 2^{6} \cdot 17^{6} \) |
$1.87867$ |
$(a+4), (a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \cdot 5 \) |
$0.019399282$ |
$11.63336164$ |
1.877761721 |
\( -\frac{487724602833}{11358856} a - \frac{614162030625}{11358856} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 49 a - 115\) , \( -82 a + 190\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(49a-115\right){x}-82a+190$ |
*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.
