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 |
3.1-a1 |
3.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{4} \) |
$2.44716$ |
$(-1202a+13107)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.382800226$ |
$33.64129478$ |
4.950977218 |
\( -\frac{897944}{81} a - \frac{8770081}{81} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -50024838009 a + 545487142956\) , \( -7185964954797827 a + 78358104659428531\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-50024838009a+545487142956\right){x}-7185964954797827a+78358104659428531$ |
3.2-a1 |
3.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{4} \) |
$2.44716$ |
$(-1202a-11905)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.382800226$ |
$33.64129478$ |
4.950977218 |
\( \frac{897944}{81} a - \frac{358075}{3} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 50024838009 a + 495462304947\) , \( 7185964954797827 a + 71172139704630704\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(50024838009a+495462304947\right){x}+7185964954797827a+71172139704630704$ |
4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{8} \) |
$2.62965$ |
$(a+10), (-a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$1$ |
$23.79940653$ |
8.006085419 |
\( \frac{653341}{128} a + \frac{1619595}{32} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -183596225915247433 a + 2001993104035084020\) , \( 738396121317306709527577766 a - 8051712041215403718971174891\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-183596225915247433a+2001993104035084020\right){x}+738396121317306709527577766a-8051712041215403718971174891$ |
4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{18} \) |
$2.62965$ |
$(a+10), (-a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3^{2} \) |
$0.224585625$ |
$27.01677809$ |
5.248607171 |
\( -\frac{1041157}{256} a - \frac{9606923}{512} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 3221641116 a - 35129825064\) , \( -307897027602031 a + 3357409570590572\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(3221641116a-35129825064\right){x}-307897027602031a+3357409570590572$ |
4.1-c1 |
4.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{18} \) |
$2.62965$ |
$(a+10), (-a+11)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3^{2} \) |
$0.224585625$ |
$27.01677809$ |
5.248607171 |
\( \frac{1041157}{256} a - \frac{11689237}{512} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -3221641117 a - 31908183948\) , \( 307897027602030 a + 3049512542988541\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-3221641117a-31908183948\right){x}+307897027602030a+3049512542988541$ |
4.1-d1 |
4.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{8} \) |
$2.62965$ |
$(a+10), (-a+11)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$1$ |
$23.79940653$ |
8.006085419 |
\( -\frac{653341}{128} a + \frac{7131721}{128} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 183596225915247432 a + 1818396878119836588\) , \( -738396121317306709527577767 a - 7313315919898097009443597124\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(183596225915247432a+1818396878119836588\right){x}-738396121317306709527577767a-7313315919898097009443597124$ |
4.1-e1 |
4.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$2.62965$ |
$(a+10), (-a+11)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$0$ |
\( 2^{2} \) |
$11.52772170$ |
$24.53113644$ |
1.286620543 |
\( -\frac{8481}{64} a - \frac{4891}{16} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -29530642419 a - 292481110197\) , \( 12559811845011234 a + 124396471304828960\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-29530642419a-292481110197\right){x}+12559811845011234a+124396471304828960$ |
4.1-e2 |
4.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$2.62965$ |
$(a+10), (-a+11)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$0$ |
\( 2^{2} \) |
$11.52772170$ |
$24.53113644$ |
1.286620543 |
\( \frac{8481}{64} a - \frac{28045}{64} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 29530642419 a - 322011752616\) , \( -12559811845011234 a + 136956283149840194\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(29530642419a-322011752616\right){x}-12559811845011234a+136956283149840194$ |
6.2-a1 |
6.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3^{4} \) |
$2.91018$ |
$(a+10), (-1202a-11905)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.170804502$ |
$13.50686390$ |
1.773903022 |
\( -\frac{3500231839}{331776} a - \frac{1281595783}{12288} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -8433891771 a - 83532013715\) , \( 1365231161498006 a + 13521694520702646\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-8433891771a-83532013715\right){x}+1365231161498006a+13521694520702646$ |
6.3-a1 |
6.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{433}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3^{4} \) |
$2.91018$ |
$(-a+11), (-1202a+13107)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.170804502$ |
$13.50686390$ |
1.773903022 |
\( \frac{3500231839}{331776} a - \frac{9525829495}{82944} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 8433891771 a - 91965905486\) , \( -1365231161498006 a + 14886925682200652\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(8433891771a-91965905486\right){x}-1365231161498006a+14886925682200652$ |
*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.