| 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-a1 |
4.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{12} \) |
$1.19220$ |
$(a+4), (a-5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$22.28205935$ |
1.049730474 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -716 a + 3738\) , \( -1996 a + 10404\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-716a+3738\right){x}-1996a+10404$ |
| 32.3-e1 |
32.3-e |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{24} \) |
$2.00503$ |
$(a+4), (a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$11.05144135$ |
4.685801761 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -1585149 a + 8269705\) , \( 70404496 a - 367299591\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-1585149a+8269705\right){x}+70404496a-367299591$ |
| 32.4-e1 |
32.4-e |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
32.4 |
\( 2^{5} \) |
\( 2^{24} \) |
$2.00503$ |
$(a+4), (a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.210288270$ |
4.685801761 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 2 a + 58\) , \( 12 a + 28\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(2a+58\right){x}+12a+28$ |
| 100.4-c1 |
100.4-c |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
100.4 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{12} \cdot 5^{6} \) |
$2.66584$ |
$(a+4), (a-5), (4a+17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2^{2} \) |
$0.425400176$ |
$9.964839881$ |
3.594702668 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -12 a + 71\) , \( -16 a + 85\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-12a+71\right){x}-16a+85$ |
| 100.5-c1 |
100.5-c |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
100.5 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{12} \cdot 5^{6} \) |
$2.66584$ |
$(a+4), (a-5), (4a-21)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.085080035$ |
$9.964839881$ |
3.594702668 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -1025297 a + 5348973\) , \( -39098117 a + 203974503\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+\left(-1025297a+5348973\right){x}-39098117a+203974503$ |
| 128.5-j1 |
128.5-j |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{30} \) |
$2.83554$ |
$(a+4), (a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.486342454$ |
$1.550342002$ |
6.393882919 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -431 a + 2313\) , \( 439 a - 2153\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-431a+2313\right){x}+439a-2153$ |
| 128.5-k1 |
128.5-k |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{30} \) |
$2.83554$ |
$(a+4), (a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2^{2} \) |
$0.523862151$ |
$7.814549122$ |
3.471490101 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -26795 a - 112984\) , \( 5145509 a + 21698572\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-26795a-112984\right){x}+5145509a+21698572$ |
| 128.6-j1 |
128.6-j |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{30} \) |
$2.83554$ |
$(a+4), (a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2^{3} \) |
$2.431712273$ |
$1.550342002$ |
6.393882919 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -74636 a + 389382\) , \( 602440 a - 3142913\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-74636a+389382\right){x}+602440a-3142913$ |
| 128.6-k1 |
128.6-k |
$4$ |
$15$ |
\(\Q(\sqrt{89}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{30} \) |
$2.83554$ |
$(a+4), (a-5)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
|
\( 2^{2} \) |
$1$ |
$1.562909824$ |
3.471490101 |
\( -\frac{818172355}{1024} a - \frac{3448457287}{1024} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -160 a - 649\) , \( -2420 a - 10179\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-160a-649\right){x}-2420a-10179$ |
*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.
