| 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 |
| 4.1-a1 |
4.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$34.81892277$ |
0.336733136 |
\( -\frac{10838595115443}{4} a + \frac{36550776099881}{4} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -746 a - 1771\) , \( 18744 a + 44466\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-746a-1771\right){x}+18744a+44466$ |
| 4.1-a2 |
4.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{27} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.868769197$ |
0.336733136 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -36 a - 86\) , \( -2492 a - 5912\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-36a-86\right){x}-2492a-5912$ |
| 4.1-a3 |
4.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \) |
$1$ |
$34.81892277$ |
0.336733136 |
\( -\frac{286425}{64} a + \frac{966617}{64} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 4 a + 9\) , \( 92 a + 218\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(4a+9\right){x}+92a+218$ |
| 4.1-a4 |
4.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \) |
$1$ |
$34.81892277$ |
0.336733136 |
\( \frac{286425}{64} a + 10628 \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -4 a + 13\) , \( -92 a + 310\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-4a+13\right){x}-92a+310$ |
| 4.1-a5 |
4.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{27} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$3.868769197$ |
0.336733136 |
\( \frac{5519537297}{262144} a + \frac{1636371017}{32768} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 36 a - 122\) , \( 2492 a - 8404\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(36a-122\right){x}+2492a-8404$ |
| 4.1-a6 |
4.1-a |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$34.81892277$ |
0.336733136 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 746 a - 2517\) , \( -18744 a + 63210\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(746a-2517\right){x}-18744a+63210$ |
| 4.1-b1 |
4.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$1.489742545$ |
1.166989006 |
\( -\frac{10838595115443}{4} a + \frac{36550776099881}{4} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 294 a - 990\) , \( 4809 a - 16223\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(294a-990\right){x}+4809a-16223$ |
| 4.1-b2 |
4.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{27} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/18\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{4} \) |
$1$ |
$13.40768290$ |
1.166989006 |
\( -\frac{5519537297}{262144} a + \frac{18610505433}{262144} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 44 a - 145\) , \( -242 a + 809\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(44a-145\right){x}-242a+809$ |
| 4.1-b3 |
4.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$13.40768290$ |
1.166989006 |
\( -\frac{286425}{64} a + \frac{966617}{64} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 4 a - 10\) , \( 7 a - 31\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-10\right){x}+7a-31$ |
| 4.1-b4 |
4.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$13.40768290$ |
1.166989006 |
\( \frac{286425}{64} a + 10628 \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -3 a - 6\) , \( -12 a - 29\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a-6\right){x}-12a-29$ |
| 4.1-b5 |
4.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{27} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/18\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{4} \) |
$1$ |
$13.40768290$ |
1.166989006 |
\( \frac{5519537297}{262144} a + \frac{1636371017}{32768} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -43 a - 101\) , \( 197 a + 467\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-43a-101\right){x}+197a+467$ |
| 4.1-b6 |
4.1-b |
$6$ |
$18$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$0.72596$ |
$(-a-2), (-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$1.489742545$ |
1.166989006 |
\( \frac{10838595115443}{4} a + \frac{12856090492219}{2} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -293 a - 696\) , \( -5104 a - 12109\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-293a-696\right){x}-5104a-12109$ |
*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.
