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 |
3.2-a1 |
3.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{16} \) |
$1.08425$ |
$(3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$36$ |
\( 2 \) |
$1$ |
$0.699459757$ |
1.365607129 |
\( -\frac{732096671152080845}{81} a + \frac{1246949079628698820}{27} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -10332 a - 42522\) , \( -1271183 a - 5224453\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-10332a-42522\right){x}-1271183a-5224453$ |
3.2-a2 |
3.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{24} \) |
$1.08425$ |
$(3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$6.295137819$ |
1.365607129 |
\( -\frac{25207270205}{531441} a + \frac{42974373355}{177147} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 198 a + 813\) , \( -7340 a - 30166\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(198a+813\right){x}-7340a-30166$ |
3.2-a3 |
3.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{18} \) |
$1.08425$ |
$(3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$25.18055127$ |
1.365607129 |
\( \frac{505895}{729} a + \frac{1119530}{243} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -127 a - 522\) , \( -2077 a - 8536\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-127a-522\right){x}-2077a-8536$ |
3.2-a4 |
3.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{14} \) |
$1.08425$ |
$(3,a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2 \) |
$1$ |
$2.797839030$ |
1.365607129 |
\( \frac{392415680105}{9} a + \frac{538445689895}{3} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -10342 a - 42507\) , \( -1270240 a - 5220400\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-10342a-42507\right){x}-1270240a-5220400$ |
3.2-b1 |
3.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{4} \) |
$1.08425$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$8.145025583$ |
$0.699459757$ |
1.235878333 |
\( -\frac{732096671152080845}{81} a + \frac{1246949079628698820}{27} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -13050 a - 53636\) , \( -1785556 a - 7338231\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-13050a-53636\right){x}-1785556a-7338231$ |
3.2-b2 |
3.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{12} \) |
$1.08425$ |
$(3,a+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$2.715008527$ |
$6.295137819$ |
1.235878333 |
\( -\frac{25207270205}{531441} a + \frac{42974373355}{177147} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 245 a + 1004\) , \( -10714 a - 44034\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(245a+1004\right){x}-10714a-44034$ |
3.2-b3 |
3.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{6} \) |
$1.08425$ |
$(3,a+2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.357504263$ |
$25.18055127$ |
1.235878333 |
\( \frac{505895}{729} a + \frac{1119530}{243} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -165 a - 681\) , \( -2743 a - 11275\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-165a-681\right){x}-2743a-11275$ |
3.2-b4 |
3.2-b |
$4$ |
$6$ |
\(\Q(\sqrt{85}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{2} \) |
$1.08425$ |
$(3,a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$4.072512791$ |
$2.797839030$ |
1.235878333 |
\( \frac{392415680105}{9} a + \frac{538445689895}{3} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -13055 a - 53656\) , \( -1784186 a - 7332600\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-13055a-53656\right){x}-1784186a-7332600$ |
*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.