| 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 |
| 3.1-a1 |
3.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{18} \) |
$1.08425$ |
$(3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$25.18055127$ |
1.365607129 |
\( -\frac{505895}{729} a + \frac{3864485}{729} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 7 a + 6\) , \( 9 a - 1\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(7a+6\right){x}+9a-1$ |
| 3.1-a2 |
3.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{24} \) |
$1.08425$ |
$(3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$6.295137819$ |
1.365607129 |
\( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -3 a + 21\) , \( -190\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-3a+21\right){x}-190$ |
| 3.1-a3 |
3.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{14} \) |
$1.08425$ |
$(3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2 \) |
$1$ |
$2.797839030$ |
1.365607129 |
\( -\frac{392415680105}{9} a + \frac{2007752749790}{9} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 177 a - 1059\) , \( 2964 a - 16948\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(177a-1059\right){x}+2964a-16948$ |
| 3.1-a4 |
3.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{16} \) |
$1.08425$ |
$(3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$36$ |
\( 2 \) |
$1$ |
$0.699459757$ |
1.365607129 |
\( \frac{732096671152080845}{81} a + \frac{3008750567734015615}{81} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -148 a - 2394\) , \( -5717 a - 52648\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-148a-2394\right){x}-5717a-52648$ |
| 3.1-b1 |
3.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$1.08425$ |
$(3,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.357504263$ |
$25.18055127$ |
1.235878333 |
\( -\frac{505895}{729} a + \frac{3864485}{729} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}+a+2$ |
| 3.1-b2 |
3.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{12} \) |
$1.08425$ |
$(3,a)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.715008527$ |
$6.295137819$ |
1.235878333 |
\( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -4 a - 19\) , \( -29 a - 121\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-4a-19\right){x}-29a-121$ |
| 3.1-b3 |
3.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{2} \) |
$1.08425$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$4.072512791$ |
$2.797839030$ |
1.235878333 |
\( -\frac{392415680105}{9} a + \frac{2007752749790}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24 a - 119\) , \( -245 a - 1051\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-24a-119\right){x}-245a-1051$ |
| 3.1-b4 |
3.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{4} \) |
$1.08425$ |
$(3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$8.145025583$ |
$0.699459757$ |
1.235878333 |
\( \frac{732096671152080845}{81} a + \frac{3008750567734015615}{81} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -434 a - 1804\) , \( -12042 a - 49534\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-434a-1804\right){x}-12042a-49534$ |
*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.
