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 |
68.1-a1 |
68.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{12} \cdot 3^{12} \cdot 17^{2} \) |
$2.36578$ |
$(17,a+8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$20.21098874$ |
6.576568561 |
\( \frac{3048625}{1088} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 31 a - 176\) , \( -80 a + 399\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(31a-176\right){x}-80a+399$ |
68.1-a2 |
68.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 3^{12} \cdot 17^{12} \) |
$2.36578$ |
$(17,a+8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.245665415$ |
6.576568561 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 1241 a - 6446\) , \( 38090 a - 194961\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1241a-6446\right){x}+38090a-194961$ |
68.1-a3 |
68.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{6} \cdot 3^{12} \cdot 17^{4} \) |
$2.36578$ |
$(17,a+8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$20.21098874$ |
6.576568561 |
\( \frac{8805624625}{2312} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 471 a - 2456\) , \( -11288 a + 57711\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(471a-2456\right){x}-11288a+57711$ |
68.1-a4 |
68.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{4} \cdot 3^{12} \cdot 17^{6} \) |
$2.36578$ |
$(17,a+8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.245665415$ |
6.576568561 |
\( \frac{120920208625}{19652} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 1131 a - 5876\) , \( 47164 a - 241377\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1131a-5876\right){x}+47164a-241377$ |
68.1-b1 |
68.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{12} \cdot 17^{2} \) |
$2.36578$ |
$(17,a+8), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.560163422$ |
$20.21098874$ |
0.818656255 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-3{x}+1$ |
68.1-b2 |
68.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{12} \) |
$2.36578$ |
$(17,a+8), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$3.360980532$ |
$2.245665415$ |
0.818656255 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \) |
${y}^2+{x}{y}={x}^{3}-113{x}-329$ |
68.1-b3 |
68.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{6} \cdot 17^{4} \) |
$2.36578$ |
$(17,a+8), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.120326844$ |
$20.21098874$ |
0.818656255 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \) |
${y}^2+{x}{y}={x}^{3}-43{x}+105$ |
68.1-b4 |
68.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{4} \cdot 17^{6} \) |
$2.36578$ |
$(17,a+8), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1.680490266$ |
$2.245665415$ |
0.818656255 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \) |
${y}^2+{x}{y}={x}^{3}-103{x}-411$ |
*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.