| 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 |
| 1058.1-a1 |
1058.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{20} \cdot 23^{2} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1.747176977$ |
0.308860172 |
\( -\frac{116930169}{23552} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -10\) , \( -12\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-10{x}-12$ |
| 1058.1-a2 |
1058.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{5} \cdot 23^{5} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.873588488$ |
0.308860172 |
\( -\frac{820801885945215895245}{2238728} a + \frac{145098644892448450536}{279841} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( 900 a - 1450\) , \( 18836 a - 27372\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(900a-1450\right){x}+18836a-27372$ |
| 1058.1-a3 |
1058.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{10} \cdot 23^{4} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$1.747176977$ |
0.308860172 |
\( \frac{545138290809}{16928} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -170\) , \( -812\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}-170{x}-812$ |
| 1058.1-a4 |
1058.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{5} \cdot 23^{5} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.873588488$ |
0.308860172 |
\( \frac{820801885945215895245}{2238728} a + \frac{145098644892448450536}{279841} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -900 a - 1450\) , \( -18836 a - 27372\bigr] \) |
${y}^2+{x}{y}={x}^{3}-{x}^{2}+\left(-900a-1450\right){x}-18836a-27372$ |
| 1058.1-b1 |
1058.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{9} \cdot 23^{3} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3^{2} \) |
$0.019941409$ |
$11.19093547$ |
2.840401669 |
\( \frac{29029935465}{16928} a - \frac{10258839081}{4232} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 10 a - 8\) , \( -18 a + 19\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(10a-8\right){x}-18a+19$ |
| 1058.1-c1 |
1058.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{9} \cdot 23^{3} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3^{2} \) |
$0.019941409$ |
$11.19093547$ |
2.840401669 |
\( -\frac{29029935465}{16928} a - \frac{10258839081}{4232} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -10 a - 8\) , \( 18 a + 19\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-10a-8\right){x}+18a+19$ |
| 1058.1-d1 |
1058.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{15} \cdot 23^{7} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.030942861$ |
$1.122257153$ |
1.636221051 |
\( -\frac{3268077984481}{37897187584} a + \frac{586928125299}{9474296896} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -18 a - 14\) , \( -166 a - 280\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-18a-14\right){x}-166a-280$ |
| 1058.1-d2 |
1058.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{5} \cdot 23^{5} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.343647620$ |
$10.10031437$ |
1.636221051 |
\( \frac{8867905}{97336} a - \frac{1710657}{24334} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 2 a + 1\) , \( 6 a + 10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(2a+1\right){x}+6a+10$ |
| 1058.1-e1 |
1058.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{5} \cdot 23^{5} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.343647620$ |
$10.10031437$ |
1.636221051 |
\( -\frac{8867905}{97336} a - \frac{1710657}{24334} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2 a + 1\) , \( -6 a + 10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-2a+1\right){x}-6a+10$ |
| 1058.1-e2 |
1058.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1058.1 |
\( 2 \cdot 23^{2} \) |
\( 2^{15} \cdot 23^{7} \) |
$1.44147$ |
$(a), (-a-5), (-a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.030942861$ |
$1.122257153$ |
1.636221051 |
\( \frac{3268077984481}{37897187584} a + \frac{586928125299}{9474296896} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 18 a - 14\) , \( 166 a - 280\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(18a-14\right){x}+166a-280$ |
*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.
