| 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 |
| 17.1-a1 |
17.1-a |
$8$ |
$16$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$8.92520$ |
$(1/2a^3-2a-3/2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.379407022$ |
$1466.529406$ |
0.992276649 |
\( \frac{5821794}{17} a^{3} - \frac{5821794}{17} a^{2} - \frac{29108970}{17} a + \frac{14936697}{17} \) |
\( \bigl[-\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{3}{2}\) , \( a^{2} - 2 a - 2\) , \( \frac{1}{2} a^{3} - 2 a - \frac{1}{2}\) , \( -\frac{5}{2} a^{3} + 4 a^{2} + 5 a - \frac{1}{2}\) , \( -2 a^{3} + 6 a^{2} - 3 a - 1\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{3}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-2a-\frac{1}{2}\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(-\frac{5}{2}a^{3}+4a^{2}+5a-\frac{1}{2}\right){x}-2a^{3}+6a^{2}-3a-1$ |
| 17.1-a2 |
17.1-a |
$8$ |
$16$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$8.92520$ |
$(1/2a^3-2a-3/2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.379407022$ |
$1466.529406$ |
0.992276649 |
\( -\frac{5821794}{17} a^{3} + \frac{5821794}{17} a^{2} + \frac{29108970}{17} a + \frac{9114903}{17} \) |
\( \bigl[-\frac{1}{2} a^{3} + a^{2} + 3 a - \frac{1}{2}\) , \( a^{3} - a^{2} - 5 a\) , \( \frac{1}{2} a^{3} - 3 a - \frac{3}{2}\) , \( 26 a^{3} - 13 a^{2} - 161 a - 51\) , \( -134 a^{3} + 69 a^{2} + 839 a + 276\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+a^{2}+3a-\frac{1}{2}\right){x}{y}+\left(\frac{1}{2}a^{3}-3a-\frac{3}{2}\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(26a^{3}-13a^{2}-161a-51\right){x}-134a^{3}+69a^{2}+839a+276$ |
| 17.1-a3 |
17.1-a |
$8$ |
$16$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$8.92520$ |
$(1/2a^3-2a-3/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$64$ |
\( 2 \) |
$1.517628088$ |
$0.358039405$ |
0.992276649 |
\( \frac{97064067741644382786}{17} a^{3} - \frac{97064067741644382786}{17} a^{2} - \frac{485320338708221913930}{17} a + \frac{248634735746274843273}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -330 a^{3} + 330 a^{2} + 1650 a - 936\) , \( -4996 a^{3} + 4996 a^{2} + 24980 a - 13110\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-330a^{3}+330a^{2}+1650a-936\right){x}-4996a^{3}+4996a^{2}+24980a-13110$ |
| 17.1-a4 |
17.1-a |
$8$ |
$16$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$8.92520$ |
$(1/2a^3-2a-3/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$64$ |
\( 2 \) |
$1.517628088$ |
$0.358039405$ |
0.992276649 |
\( -\frac{97064067741644382786}{17} a^{3} + \frac{97064067741644382786}{17} a^{2} + \frac{485320338708221913930}{17} a + \frac{151570668004630460487}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 330 a^{3} - 330 a^{2} - 1650 a - 606\) , \( 4996 a^{3} - 4996 a^{2} - 24980 a - 8114\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(330a^{3}-330a^{2}-1650a-606\right){x}+4996a^{3}-4996a^{2}-24980a-8114$ |
| 17.1-a5 |
17.1-a |
$8$ |
$16$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$8.92520$ |
$(1/2a^3-2a-3/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2^{2} \) |
$0.758814044$ |
$5.728630493$ |
0.992276649 |
\( \frac{82483294977}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -91\) , \( -310\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-91{x}-310$ |
| 17.1-a6 |
17.1-a |
$8$ |
$16$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{8} \) |
$8.92520$ |
$(1/2a^3-2a-3/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$0.379407022$ |
$91.65808789$ |
0.992276649 |
\( \frac{20346417}{289} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -6\) , \( -4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-6{x}-4$ |
| 17.1-a7 |
17.1-a |
$8$ |
$16$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{16} \) |
$8.92520$ |
$(1/2a^3-2a-3/2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$0.758814044$ |
$5.728630493$ |
0.992276649 |
\( -\frac{35937}{83521} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( -14\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}-14$ |
| 17.1-a8 |
17.1-a |
$8$ |
$16$ |
4.4.4913.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$8.92520$ |
$(1/2a^3-2a-3/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.758814044$ |
$1466.529406$ |
0.992276649 |
\( \frac{35937}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}$ |
*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.
