| 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 |
| 68.1-a1 |
68.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( - 2^{5} \cdot 17^{3} \) |
$1.05801$ |
$(-a+2), (-a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$1.122832707$ |
2.450942393 |
\( -\frac{42449501347685375}{4624} a + \frac{27184159892778875}{1156} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -611 a - 955\) , \( -16227 a - 25345\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-611a-955\right){x}-16227a-25345$ |
| 68.1-a2 |
68.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( - 2^{15} \cdot 17 \) |
$1.05801$ |
$(-a+2), (-a-1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$10.10549437$ |
2.450942393 |
\( -\frac{562364375}{69632} a + \frac{378001125}{17408} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 50 a - 128\) , \( 290 a - 743\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(50a-128\right){x}+290a-743$ |
| 68.1-a3 |
68.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{12} \cdot 17^{2} \) |
$1.05801$ |
$(-a+2), (-a-1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$20.21098874$ |
2.450942393 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-3{x}+1$ |
| 68.1-a4 |
68.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{12} \) |
$1.05801$ |
$(-a+2), (-a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$2.245665415$ |
2.450942393 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \) |
${y}^2+{x}{y}={x}^{3}-113{x}-329$ |
| 68.1-a5 |
68.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( - 2^{15} \cdot 17 \) |
$1.05801$ |
$(-a+2), (-a-1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$10.10549437$ |
2.450942393 |
\( \frac{562364375}{69632} a + \frac{949640125}{69632} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -50 a - 78\) , \( -290 a - 453\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-50a-78\right){x}-290a-453$ |
| 68.1-a6 |
68.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{6} \cdot 17^{4} \) |
$1.05801$ |
$(-a+2), (-a-1), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$20.21098874$ |
2.450942393 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \) |
${y}^2+{x}{y}={x}^{3}-43{x}+105$ |
| 68.1-a7 |
68.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{4} \cdot 17^{6} \) |
$1.05801$ |
$(-a+2), (-a-1), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{3} \) |
$1$ |
$2.245665415$ |
2.450942393 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \) |
${y}^2+{x}{y}={x}^{3}-103{x}-411$ |
| 68.1-a8 |
68.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( - 2^{5} \cdot 17^{3} \) |
$1.05801$ |
$(-a+2), (-a-1), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$1.122832707$ |
2.450942393 |
\( \frac{42449501347685375}{4624} a + \frac{66287138223430125}{4624} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 610 a - 1565\) , \( 16226 a - 41571\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(610a-1565\right){x}+16226a-41571$ |
*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.
