| 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 |
| 576.1-a1 |
576.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( - 2^{21} \cdot 3^{8} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.462620034$ |
1.679617428 |
\( -\frac{1196518}{27} a + \frac{9334094}{81} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 144 a + 224\) , \( 144 a + 224\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(144a+224\right){x}+144a+224$ |
| 576.1-a2 |
576.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{4} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$13.85048013$ |
1.679617428 |
\( -\frac{52}{9} a + \frac{15764}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -36 a - 56\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-36a-56\right){x}$ |
| 576.1-a3 |
576.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{2} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.85048013$ |
1.679617428 |
\( -\frac{165886}{3} a + \frac{1108246}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( a - 12\) , \( -8 a + 12\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(a-12\right){x}-8a+12$ |
| 576.1-a4 |
576.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( - 2^{18} \cdot 3^{2} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$13.85048013$ |
1.679617428 |
\( \frac{264004}{3} a + \frac{417344}{3} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -3 a + 8\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a+8\right){x}$ |
| 576.1-b1 |
576.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( - 2^{18} \cdot 3^{2} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$13.85048013$ |
1.679617428 |
\( -\frac{264004}{3} a + 227116 \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 5 a + 4\) , \( -4 a - 4\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(5a+4\right){x}-4a-4$ |
| 576.1-b2 |
576.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{4} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$13.85048013$ |
1.679617428 |
\( \frac{52}{9} a + \frac{15712}{9} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 36 a - 92\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(36a-92\right){x}$ |
| 576.1-b3 |
576.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( - 2^{21} \cdot 3^{8} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.462620034$ |
1.679617428 |
\( \frac{1196518}{27} a + \frac{5744540}{81} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -144 a + 368\) , \( -144 a + 368\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-144a+368\right){x}-144a+368$ |
| 576.1-b4 |
576.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{15} \cdot 3^{2} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.85048013$ |
1.679617428 |
\( \frac{165886}{3} a + 314120 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -a - 11\) , \( 8 a + 4\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-a-11\right){x}+8a+4$ |
| 576.1-c1 |
576.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{16} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.540027220$ |
$1.162639934$ |
2.864963790 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 16\) , \( -180\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+16{x}-180$ |
| 576.1-c2 |
576.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.270013610$ |
$18.60223895$ |
2.864963790 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+{x}$ |
| 576.1-c3 |
576.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{4} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.635006805$ |
$18.60223895$ |
2.864963790 |
\( \frac{35152}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -4\) , \( 4\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-4{x}+4$ |
| 576.1-c4 |
576.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{8} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.270013610$ |
$4.650559737$ |
2.864963790 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -24\) , \( -36\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-24{x}-36$ |
| 576.1-c5 |
576.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{2} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.270013610$ |
$18.60223895$ |
2.864963790 |
\( \frac{28756228}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -64\) , \( 220\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-64{x}+220$ |
| 576.1-c6 |
576.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
576.1 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{22} \cdot 3^{4} \) |
$1.80496$ |
$(-a+2), (-a-1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$2.540027220$ |
$1.162639934$ |
2.864963790 |
\( \frac{3065617154}{9} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -384\) , \( -2772\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-384{x}-2772$ |
*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.
