| 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-a5 |
4.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$0.52105$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$45.92271576$ |
0.309385960 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -2 a\) , \( 0\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}-2a{x}$ |
| 32.3-a5 |
32.3-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{18} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$23.31469860$ |
1.413661249 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -2\) , \( -4 a - 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}-2{x}-4a-5$ |
| 32.4-a5 |
32.4-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.4 |
\( 2^{5} \) |
\( 2^{18} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$11.65734930$ |
1.413661249 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -7 a - 12\) , \( -7 a - 12\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-7a-12\right){x}-7a-12$ |
| 128.5-b5 |
128.5-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{24} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$8.369837957$ |
1.014991940 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -33 a - 52\) , \( 13 a + 20\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-33a-52\right){x}+13a+20$ |
| 128.5-c5 |
128.5-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{24} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$8.242990742$ |
1.999218911 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 9 a - 24\) , \( 15 a - 41\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(9a-24\right){x}+15a-41$ |
| 128.6-b5 |
128.6-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{24} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$8.369837957$ |
1.014991940 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -9\) , \( -a - 7\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}-9{x}-a-7$ |
| 128.6-c5 |
128.6-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{24} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$16.48598148$ |
1.999218911 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -146 a - 228\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-146a-228\right){x}$ |
| 256.1-b5 |
256.1-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{30} \) |
$1.47375$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$1$ |
$5.918369177$ |
1.435415367 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -15 a - 28\) , \( -16 a - 28\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-15a-28\right){x}-16a-28$ |
| 324.1-e5 |
324.1-e |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{6} \cdot 3^{12} \) |
$1.56315$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{5} \) |
$1$ |
$7.891158903$ |
3.827774313 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -9 a - 17\) , \( 4 a + 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-9a-17\right){x}+4a+5$ |
| 676.4-i5 |
676.4-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.4 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (-2a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{4} \) |
$0.989827765$ |
$6.565841088$ |
3.152503187 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -319 a - 497\) , \( 124 a + 193\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-319a-497\right){x}+124a+193$ |
| 676.5-i5 |
676.5-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.5 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{6} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{5} \) |
$0.494913882$ |
$6.565841088$ |
3.152503187 |
\( -\frac{915957}{16} a + \frac{2374013}{16} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 20 a - 55\) , \( 77 a - 199\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(20a-55\right){x}+77a-199$ |
*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.
