| 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-a6 |
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 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 981 a - 2517\) , \( 23628 a - 60528\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(981a-2517\right){x}+23628a-60528$ |
| 32.3-a6 |
32.3-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{21} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$0.971445775$ |
1.413661249 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 6440 a - 16514\) , \( 410032 a - 1050341\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(6440a-16514\right){x}+410032a-1050341$ |
| 32.4-a6 |
32.4-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.4 |
\( 2^{5} \) |
\( 2^{21} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.771566201$ |
1.413661249 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 2388 a - 6147\) , \( -92365 a + 236610\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(2388a-6147\right){x}-92365a+236610$ |
| 128.5-b6 |
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 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 1996 a - 5262\) , \( -72821 a + 185664\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(1996a-5262\right){x}-72821a+185664$ |
| 128.5-c6 |
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 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$5.495327161$ |
1.999218911 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -2431 a - 3923\) , \( 73584 a + 115495\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-2431a-3923\right){x}+73584a+115495$ |
| 128.6-b6 |
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 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -10853 a - 16976\) , \( -715483 a - 1117196\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-10853a-16976\right){x}-715483a-1117196$ |
| 128.6-c6 |
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 |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.686915895$ |
1.999218911 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 298 a - 1433\) , \( 6138 a - 22281\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(298a-1433\right){x}+6138a-22281$ |
| 256.1-b6 |
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 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 15697 a - 40284\) , \( -1536816 a + 3936512\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(15697a-40284\right){x}-1536816a+3936512$ |
| 324.1-e6 |
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 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 8829 a - 22661\) , \( -651802 a + 1669545\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(8829a-22661\right){x}-651802a+1669545$ |
| 676.4-i6 |
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 |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.329942588$ |
$3.282920544$ |
3.152503187 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 199 a - 1990\) , \( -14489 a + 19668\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(199a-1990\right){x}-14489a+19668$ |
| 676.5-i6 |
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 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.659885177$ |
$3.282920544$ |
3.152503187 |
\( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -2916 a - 4828\) , \( -99831 a - 154379\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2916a-4828\right){x}-99831a-154379$ |
*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.
