| 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 |
| 1225.1-a1 |
1225.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{4} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$8.640496728$ |
5.656532900 |
\( -\frac{9441441}{125} a - \frac{16862229}{125} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -41 a - 72\) , \( 153 a + 275\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-41a-72\right){x}+153a+275$ |
| 1225.1-a2 |
1225.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{4} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$8.640496728$ |
5.656532900 |
\( \frac{9441441}{125} a - \frac{5260734}{25} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 44 a - 110\) , \( -223 a + 640\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(44a-110\right){x}-223a+640$ |
| 1225.1-b1 |
1225.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.545174856$ |
$4.882829994$ |
4.647161457 |
\( -\frac{4882432}{625} a - \frac{8081408}{625} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -2 a + 4\) , \( -2 a + 5\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a+4\right){x}-2a+5$ |
| 1225.1-c1 |
1225.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Ns |
$1$ |
\( 2 \) |
$3.273334507$ |
$1.975948723$ |
5.645681880 |
\( -\frac{110592}{125} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 35 a - 98\) , \( 294 a - 821\bigr] \) |
${y}^2+{y}={x}^{3}+\left(35a-98\right){x}+294a-821$ |
| 1225.1-d1 |
1225.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.545174856$ |
$4.882829994$ |
4.647161457 |
\( \frac{4882432}{625} a - \frac{2592768}{125} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 2 a + 2\) , \( 2 a + 3\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(2a+2\right){x}+2a+3$ |
| 1225.1-e1 |
1225.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{10} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 5 \) |
$1.232696358$ |
$0.909163111$ |
4.891232056 |
\( \frac{801142366208}{153125} a - \frac{447243538432}{30625} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 35 a - 65\) , \( 357 a + 39\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(35a-65\right){x}+357a+39$ |
| 1225.1-f1 |
1225.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( - 5^{15} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.003875494$ |
2.628763109 |
\( -\frac{394236075318246}{244140625} a + \frac{1100615371055451}{244140625} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 860 a - 2373\) , \( 20626 a - 57467\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(860a-2373\right){x}+20626a-57467$ |
| 1225.1-f2 |
1225.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{12} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$4.015501978$ |
2.628763109 |
\( \frac{55306341}{15625} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 55 a - 168\) , \( 263 a - 739\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(55a-168\right){x}+263a-739$ |
| 1225.1-f3 |
1225.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$8.031003956$ |
2.628763109 |
\( \frac{2803221}{125} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 20 a - 63\) , \( -80 a + 213\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(20a-63\right){x}-80a+213$ |
| 1225.1-f4 |
1225.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( - 5^{15} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.007750989$ |
2.628763109 |
\( \frac{394236075318246}{244140625} a + \frac{141275859147441}{48828125} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -190 a + 357\) , \( 1852 a - 4519\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-190a+357\right){x}+1852a-4519$ |
| 1225.1-g1 |
1225.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{10} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.928724086$ |
$3.030510673$ |
2.456704215 |
\( -\frac{801142366208}{153125} a - \frac{1435075325952}{153125} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -163 a + 452\) , \( 24697 a - 68936\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-163a+452\right){x}+24697a-68936$ |
| 1225.1-h1 |
1225.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{10} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 5 \) |
$1.232696358$ |
$0.909163111$ |
4.891232056 |
\( -\frac{801142366208}{153125} a - \frac{1435075325952}{153125} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -35 a - 30\) , \( -357 a + 396\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(-35a-30\right){x}-357a+396$ |
| 1225.1-i1 |
1225.1-i |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( - 5^{15} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$2.007750989$ |
2.628763109 |
\( -\frac{394236075318246}{244140625} a + \frac{1100615371055451}{244140625} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 188 a + 169\) , \( -1853 a - 2666\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(188a+169\right){x}-1853a-2666$ |
| 1225.1-i2 |
1225.1-i |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{12} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2Cs, 3Ns |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$4.015501978$ |
2.628763109 |
\( \frac{55306341}{15625} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -57 a - 111\) , \( -264 a - 475\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-57a-111\right){x}-264a-475$ |
| 1225.1-i3 |
1225.1-i |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$8.031003956$ |
2.628763109 |
\( \frac{2803221}{125} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -22 a - 41\) , \( 79 a + 134\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-22a-41\right){x}+79a+134$ |
| 1225.1-i4 |
1225.1-i |
$4$ |
$4$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( - 5^{15} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.003875494$ |
2.628763109 |
\( \frac{394236075318246}{244140625} a + \frac{141275859147441}{48828125} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -862 a - 1511\) , \( -20627 a - 36840\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-862a-1511\right){x}-20627a-36840$ |
| 1225.1-j1 |
1225.1-j |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{10} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.928724086$ |
$3.030510673$ |
2.456704215 |
\( \frac{801142366208}{153125} a - \frac{447243538432}{30625} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 163 a + 289\) , \( -24697 a - 44239\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(163a+289\right){x}-24697a-44239$ |
| 1225.1-k1 |
1225.1-k |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$0.061898400$ |
$13.58775316$ |
2.936550080 |
\( \frac{4882432}{625} a - \frac{2592768}{125} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 13 a - 35\) , \( -53 a + 148\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(13a-35\right){x}-53a+148$ |
| 1225.1-l1 |
1225.1-l |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$3$ |
3Ns |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.032236886$ |
$10.72190660$ |
2.715300910 |
\( -\frac{110592}{125} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -7\) , \( 12\bigr] \) |
${y}^2+{y}={x}^{3}-7{x}+12$ |
| 1225.1-m1 |
1225.1-m |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$0.061898400$ |
$13.58775316$ |
2.936550080 |
\( -\frac{4882432}{625} a - \frac{8081408}{625} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -13 a - 22\) , \( 53 a + 95\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-13a-22\right){x}+53a+95$ |
| 1225.1-n1 |
1225.1-n |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{4} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$1.572392390$ |
0.343124150 |
\( -\frac{9441441}{125} a - \frac{16862229}{125} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -8 a + 27\) , \( 40 a - 128\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-8a+27\right){x}+40a-128$ |
| 1225.1-n2 |
1225.1-n |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.1 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{4} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (-a+1), (a+3)$ |
$0$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$1.572392390$ |
0.343124150 |
\( \frac{9441441}{125} a - \frac{5260734}{25} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 12 a + 12\) , \( -25 a - 52\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(12a+12\right){x}-25a-52$ |
*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.
