| 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 |
| 12.1-a1 |
12.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{20} \cdot 3^{8} \) |
$1.05187$ |
$(2,a), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$19.19785289$ |
1.517723533 |
\( -\frac{1957662404}{6561} a + \frac{6190492076}{6561} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -14 a - 42\) , \( 390 a + 1238\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-14a-42\right){x}+390a+1238$ |
| 12.1-a2 |
12.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$1.05187$ |
$(2,a), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$19.19785289$ |
1.517723533 |
\( -\frac{77824}{9} a + \frac{262144}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -4 a - 7\) , \( -3 a - 12\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a-7\right){x}-3a-12$ |
| 12.1-a3 |
12.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{16} \cdot 3^{4} \) |
$1.05187$ |
$(2,a), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$38.39570578$ |
1.517723533 |
\( \frac{2053600}{81} a + \frac{6678128}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -34 a - 102\) , \( 162 a + 510\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-34a-102\right){x}+162a+510$ |
| 12.1-a4 |
12.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{20} \cdot 3^{2} \) |
$1.05187$ |
$(2,a), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$19.19785289$ |
1.517723533 |
\( \frac{37287936100}{9} a + \frac{117914819012}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -534 a - 1682\) , \( 11534 a + 36470\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-534a-1682\right){x}+11534a+36470$ |
| 12.1-b1 |
12.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{8} \cdot 3^{8} \) |
$1.05187$ |
$(2,a), (3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.131320837$ |
$19.19785289$ |
1.195852355 |
\( -\frac{1957662404}{6561} a + \frac{6190492076}{6561} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -5 a - 11\) , \( 50 a + 161\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-5a-11\right){x}+50a+161$ |
| 12.1-b2 |
12.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$1.05187$ |
$(2,a), (3,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.131320837$ |
$19.19785289$ |
1.195852355 |
\( -\frac{77824}{9} a + \frac{262144}{9} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -14 a - 39\) , \( a + 5\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-14a-39\right){x}+a+5$ |
| 12.1-b3 |
12.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$1.05187$ |
$(2,a), (3,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 3 \) |
$0.262641675$ |
$38.39570578$ |
1.195852355 |
\( \frac{2053600}{81} a + \frac{6678128}{81} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -10 a - 26\) , \( 24 a + 80\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-10a-26\right){x}+24a+80$ |
| 12.1-b4 |
12.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{10}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{8} \cdot 3^{2} \) |
$1.05187$ |
$(2,a), (3,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.525283350$ |
$19.19785289$ |
1.195852355 |
\( \frac{37287936100}{9} a + \frac{117914819012}{9} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -135 a - 421\) , \( 1518 a + 4805\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-135a-421\right){x}+1518a+4805$ |
*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.
