| 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-a9 |
4.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{3} \) |
$0.52105$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$22.96135788$ |
0.309385960 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -12 a - 20\) , \( -28 a - 44\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-12a-20\right){x}-28a-44$ |
| 32.3-a9 |
32.3-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{15} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$23.31469860$ |
1.413661249 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -20 a - 62\) , \( 44 a + 139\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-20a-62\right){x}+44a+139$ |
| 32.4-a9 |
32.4-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.4 |
\( 2^{5} \) |
\( 2^{15} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$2.914337325$ |
1.413661249 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -77 a - 132\) , \( -619 a - 988\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-77a-132\right){x}-619a-988$ |
| 128.5-b9 |
128.5-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.184918978$ |
1.014991940 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -373 a - 592\) , \( 5721 a + 8920\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-373a-592\right){x}+5721a+8920$ |
| 128.5-c9 |
128.5-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$2.060747685$ |
1.999218911 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 149 a - 404\) , \( 1359 a - 3561\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(149a-404\right){x}+1359a-3561$ |
| 128.6-b9 |
128.6-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.184918978$ |
1.014991940 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 10 a - 129\) , \( 211 a - 247\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(10a-129\right){x}+211a-247$ |
| 128.6-c9 |
128.6-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$16.48598148$ |
1.999218911 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -1656 a - 2588\) , \( 51666 a + 80680\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-1656a-2588\right){x}+51666a+80680$ |
| 256.1-b9 |
256.1-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{27} \) |
$1.47375$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.959184588$ |
1.435415367 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -175 a - 348\) , \( 1936 a + 2788\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-175a-348\right){x}+1936a+2788$ |
| 324.1-e9 |
324.1-e |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{3} \cdot 3^{12} \) |
$1.56315$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2^{2} \) |
$1$ |
$3.945579451$ |
3.827774313 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -99 a - 197\) , \( 940 a + 1373\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-99a-197\right){x}+940a+1373$ |
| 676.4-i9 |
676.4-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.4 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{3} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1.979655531$ |
$3.282920544$ |
3.152503187 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 2806 a - 7189\) , \( 114963 a - 294484\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(2806a-7189\right){x}+114963a-294484$ |
| 676.5-i9 |
676.5-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.5 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{3} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.989827765$ |
$3.282920544$ |
3.152503187 |
\( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 330 a - 875\) , \( 4885 a - 12459\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(330a-875\right){x}+4885a-12459$ |
*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.
