| 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-a3 |
4.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{27} \) |
$0.52105$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.551261986$ |
0.309385960 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 9 a + 13\) , \( -106 a - 166\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(9a+13\right){x}-106a-166$ |
| 32.3-a3 |
32.3-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{39} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1.942891550$ |
1.413661249 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 21 a + 29\) , \( 404 a + 627\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(21a+29\right){x}+404a+627$ |
| 32.4-a3 |
32.4-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.4 |
\( 2^{5} \) |
\( 2^{39} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$3.885783100$ |
1.413661249 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 58 a + 88\) , \( -1783 a - 2784\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(58a+88\right){x}-1783a-2784$ |
| 128.5-b3 |
128.5-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{45} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.184918978$ |
1.014991940 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 274 a + 429\) , \( 18518 a + 28917\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(274a+429\right){x}+18518a+28917$ |
| 128.5-c3 |
128.5-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{45} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.747663580$ |
1.999218911 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 24 a - 41\) , \( -98 a + 129\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(24a-41\right){x}-98a+129$ |
| 128.6-b3 |
128.6-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{45} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.184918978$ |
1.014991940 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 23 a + 19\) , \( 329 a + 535\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(23a+19\right){x}+329a+535$ |
| 128.6-c3 |
128.6-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{45} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$1.373831790$ |
1.999218911 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 285 a - 728\) , \( 4041 a - 10348\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(285a-728\right){x}+4041a-10348$ |
| 256.1-b3 |
256.1-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{51} \) |
$1.47375$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.959184588$ |
1.435415367 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 144 a + 216\) , \( 6784 a + 10608\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(144a+216\right){x}+6784a+10608$ |
| 324.1-e3 |
324.1-e |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{27} \cdot 3^{12} \) |
$1.56315$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1$ |
$3.945579451$ |
3.827774313 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 81 a + 121\) , \( 2862 a + 4475\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(81a+121\right){x}+2862a+4475$ |
| 676.4-i3 |
676.4-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.4 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{27} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{5} \cdot 3 \) |
$0.082485647$ |
$3.282920544$ |
3.152503187 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 345 a - 882\) , \( -5169 a + 13248\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(345a-882\right){x}-5169a+13248$ |
| 676.5-i3 |
676.5-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.5 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{27} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.659885177$ |
$3.282920544$ |
3.152503187 |
\( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 47 a - 97\) , \( -189 a + 693\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(47a-97\right){x}-189a+693$ |
*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.
