| 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-a12 |
4.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{9} \) |
$0.52105$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.551261986$ |
0.309385960 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -982 a - 1535\) , \( -23629 a - 36899\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-982a-1535\right){x}-23629a-36899$ |
| 32.3-a12 |
32.3-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{21} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.771566201$ |
1.413661249 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -2385 a - 3757\) , \( 86220 a + 134699\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2385a-3757\right){x}+86220a+134699$ |
| 32.4-a12 |
32.4-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.4 |
\( 2^{5} \) |
\( 2^{21} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$0.971445775$ |
1.413661249 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -6442 a - 10072\) , \( -410033 a - 640308\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-6442a-10072\right){x}-410033a-640308$ |
| 128.5-b12 |
128.5-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{27} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.184918978$ |
1.014991940 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 10858 a - 27827\) , \( 698508 a - 1789259\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(10858a-27827\right){x}+698508a-1789259$ |
| 128.5-c12 |
128.5-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{27} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.686915895$ |
1.999218911 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -296 a - 1139\) , \( -7277 a - 16189\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-296a-1139\right){x}-7277a-16189$ |
| 128.6-b12 |
128.6-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{27} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.184918978$ |
1.014991940 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -1995 a - 3269\) , \( 69553 a + 108129\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-1995a-3269\right){x}+69553a+108129$ |
| 128.6-c12 |
128.6-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{27} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$5.495327161$ |
1.999218911 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 2429 a - 6352\) , \( -73585 a + 189080\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(2429a-6352\right){x}-73585a+189080$ |
| 256.1-b12 |
256.1-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{33} \) |
$1.47375$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.959184588$ |
1.435415367 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -15695 a - 24588\) , \( 1521120 a + 2375108\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-15695a-24588\right){x}+1521120a+2375108$ |
| 324.1-e12 |
324.1-e |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{9} \cdot 3^{12} \) |
$1.56315$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.945579451$ |
3.827774313 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -8829 a - 13832\) , \( 651802 a + 1017743\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-8829a-13832\right){x}+651802a+1017743$ |
| 676.4-i12 |
676.4-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.4 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{9} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.659885177$ |
$3.282920544$ |
3.152503187 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 2916 a - 7744\) , \( 99831 a - 254210\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(2916a-7744\right){x}+99831a-254210$ |
| 676.5-i12 |
676.5-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.5 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{9} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.329942588$ |
$3.282920544$ |
3.152503187 |
\( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -200 a - 1790\) , \( 14488 a + 5180\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-200a-1790\right){x}+14488a+5180$ |
*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.
