| 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 |
| 225.1-a1 |
225.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{2} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.1.1[2] |
$1$ |
\( 1 \) |
$1$ |
$48.75642491$ |
0.872181443 |
\( -\frac{102400}{3} \) |
\( \bigl[0\) , \( -\phi + 1\) , \( 1\) , \( -2 \phi - 1\) , \( 2 \phi + 1\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-2\phi-1\right){x}+2\phi+1$ |
| 225.1-a2 |
225.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{10} \cdot 5^{10} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.1.4[2] |
$1$ |
\( 1 \) |
$1$ |
$1.950256996$ |
0.872181443 |
\( \frac{20480}{243} \) |
\( \bigl[0\) , \( \phi\) , \( 1\) , \( -8 \phi + 17\) , \( 76 \phi - 131\bigr] \) |
${y}^2+{y}={x}^{3}+\phi{x}^{2}+\left(-8\phi+17\right){x}+76\phi-131$ |
| 225.1-b1 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( - 3^{4} \cdot 5^{7} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.278990851$ |
1.019195692 |
\( -\frac{152409672113485069453847362}{45} a + \frac{246604029693845863366701161}{45} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( 16875 \phi - 55575\) , \( -2120029 \phi + 5229447\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(16875\phi-55575\right){x}-2120029\phi+5229447$ |
| 225.1-b2 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{32} \cdot 5^{8} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.139495425$ |
1.019195692 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( -550 \phi - 550\) , \( 15946 \phi + 12097\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(-550\phi-550\right){x}+15946\phi+12097$ |
| 225.1-b3 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{8} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$4.557981702$ |
1.019195692 |
\( -\frac{1}{15} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( 0\) , \( -4 \phi - 3\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}-4\phi-3$ |
| 225.1-b4 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{4} \cdot 5^{22} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.139495425$ |
1.019195692 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( 175 \phi + 175\) , \( 1081 \phi + 767\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(175\phi+175\right){x}+1081\phi+767$ |
| 225.1-b5 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{8} \cdot 5^{14} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$4.557981702$ |
1.019195692 |
\( \frac{111284641}{50625} \) |
\( \bigl[\phi\) , \( \phi - 1\) , \( \phi + 1\) , \( 49 \phi - 100\) , \( -147 \phi + 243\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(49\phi-100\right){x}-147\phi+243$ |
| 225.1-b6 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{4} \cdot 5^{10} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$4.557981702$ |
1.019195692 |
\( \frac{13997521}{225} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( -25 \phi - 25\) , \( -119 \phi - 83\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(-25\phi-25\right){x}-119\phi-83$ |
| 225.1-b7 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{16} \cdot 5^{10} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$4.557981702$ |
1.019195692 |
\( \frac{272223782641}{164025} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( -675 \phi - 675\) , \( 11171 \phi + 8547\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(-675\phi-675\right){x}+11171\phi+8547$ |
| 225.1-b8 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{8} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$1.139495425$ |
1.019195692 |
\( \frac{56667352321}{15} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( -400 \phi - 400\) , \( -6044 \phi - 4433\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(-400\phi-400\right){x}-6044\phi-4433$ |
| 225.1-b9 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{8} \cdot 5^{8} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$4.557981702$ |
1.019195692 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi + 1\) , \( -10800 \phi - 10800\) , \( 758396 \phi + 571497\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(-10800\phi-10800\right){x}+758396\phi+571497$ |
| 225.1-b10 |
225.1-b |
$10$ |
$32$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( - 3^{4} \cdot 5^{7} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.278990851$ |
1.019195692 |
\( \frac{152409672113485069453847362}{45} a + \frac{94194357580360793912853799}{45} \) |
\( \bigl[\phi\) , \( \phi - 1\) , \( \phi + 1\) , \( -16876 \phi - 38700\) , \( 2081328 \phi + 3131243\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-16876\phi-38700\right){x}+2081328\phi+3131243$ |
| 225.1-c1 |
225.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{2} \cdot 5^{8} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.1.4[2] |
$1$ |
\( 1 \) |
$1$ |
$1.967118283$ |
0.879722040 |
\( -\frac{102400}{3} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -8\) , \( -7\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}-8{x}-7$ |
| 225.1-c2 |
225.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
225.1 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{10} \cdot 5^{4} \) |
$0.77387$ |
$(-2a+1), (3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$5$ |
5B.1.1[2] |
$1$ |
\( 5 \) |
$1$ |
$9.835591419$ |
0.879722040 |
\( \frac{20480}{243} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 2\) , \( 4\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+2{x}+4$ |
*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.
