| 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 |
| 3362.1-a1 |
3362.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3362.1 |
\( 2 \cdot 41^{2} \) |
\( - 2^{4} \cdot 41^{8} \) |
$1.92457$ |
$(a), (-2a+7), (2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.187804837$ |
$2.615921646$ |
4.168672289 |
\( \frac{14356103997325}{19000416964} a + \frac{41326572582217}{19000416964} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -45 a - 27\) , \( 88 a + 61\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-45a-27\right){x}+88a+61$ |
| 3362.1-a2 |
3362.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3362.1 |
\( 2 \cdot 41^{2} \) |
\( 2^{8} \cdot 41^{4} \) |
$1.92457$ |
$(a), (-2a+7), (2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.093902418$ |
$10.46368658$ |
4.168672289 |
\( \frac{1135868161485}{551368} a + \frac{3215614924651}{1102736} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -25 a - 47\) , \( 92 a + 121\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-25a-47\right){x}+92a+121$ |
| 3362.1-b1 |
3362.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3362.1 |
\( 2 \cdot 41^{2} \) |
\( - 2^{4} \cdot 41^{8} \) |
$1.92457$ |
$(a), (-2a+7), (2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.187804837$ |
$2.615921646$ |
4.168672289 |
\( -\frac{14356103997325}{19000416964} a + \frac{41326572582217}{19000416964} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 45 a - 27\) , \( -88 a + 61\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(45a-27\right){x}-88a+61$ |
| 3362.1-b2 |
3362.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3362.1 |
\( 2 \cdot 41^{2} \) |
\( 2^{8} \cdot 41^{4} \) |
$1.92457$ |
$(a), (-2a+7), (2a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.093902418$ |
$10.46368658$ |
4.168672289 |
\( -\frac{1135868161485}{551368} a + \frac{3215614924651}{1102736} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 25 a - 47\) , \( -92 a + 121\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(25a-47\right){x}-92a+121$ |
| 3362.1-c1 |
3362.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3362.1 |
\( 2 \cdot 41^{2} \) |
\( - 2 \cdot 41^{5} \) |
$1.92457$ |
$(a), (-2a+7), (2a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.120199585$ |
$1.682854345$ |
2.522947863 |
\( -\frac{4906399929094573}{5651522} a + \frac{3469348661100520}{2825761} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 6 a + 3\) , \( -3071 a - 4348\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(6a+3\right){x}-3071a-4348$ |
| 3362.1-c2 |
3362.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3362.1 |
\( 2 \cdot 41^{2} \) |
\( 2^{4} \cdot 41^{2} \) |
$1.92457$ |
$(a), (-2a+7), (2a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.132512474$ |
$26.92566952$ |
2.522947863 |
\( \frac{389017}{164} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-2{x}$ |
| 3362.1-c3 |
3362.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3362.1 |
\( 2 \cdot 41^{2} \) |
\( 2^{2} \cdot 41^{4} \) |
$1.92457$ |
$(a), (-2a+7), (2a+7)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.530049896$ |
$6.731417382$ |
2.522947863 |
\( \frac{169112377}{3362} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -12\) , \( -16\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-12{x}-16$ |
| 3362.1-c4 |
3362.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
3362.1 |
\( 2 \cdot 41^{2} \) |
\( - 2 \cdot 41^{5} \) |
$1.92457$ |
$(a), (-2a+7), (2a+7)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.120199585$ |
$1.682854345$ |
2.522947863 |
\( \frac{4906399929094573}{5651522} a + \frac{3469348661100520}{2825761} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -8 a + 3\) , \( 3070 a - 4348\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-8a+3\right){x}+3070a-4348$ |
*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.
