| 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 |
| 1225.2-a1 |
1225.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{8} \cdot 7^{4} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 1 \) |
$1.248524248$ |
$5.021791350$ |
2.736377394 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( a + 2\) , \( -182 a + 508\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}-182a+508$ |
| 1225.2-a2 |
1225.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{8} \cdot 7^{4} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.2.1 |
$1$ |
\( 3 \) |
$0.416174749$ |
$5.021791350$ |
2.736377394 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( a + 2\) , \( 3 a - 3\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}+3a-3$ |
| 1225.2-b1 |
1225.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{2} \cdot 7^{10} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 1 \) |
$1.382361320$ |
$7.351149817$ |
4.435036470 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( a + 2\) , \( 17 a + 32\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}+17a+32$ |
| 1225.2-b2 |
1225.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{2} \cdot 7^{10} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 1 \) |
$4.147083961$ |
$2.450383272$ |
4.435036470 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( a + 2\) , \( 56 a - 157\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}+56a-157$ |
| 1225.2-c1 |
1225.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Cs.6.2 |
$1$ |
\( 2 \) |
$0.156143787$ |
$15.73127654$ |
2.144070295 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( 1\) , \( a + 2\) , \( -18 a + 48\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}-18a+48$ |
| 1225.2-c2 |
1225.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{6} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Cs.6.2 |
$1$ |
\( 2 \) |
$0.468431363$ |
$5.243758848$ |
2.144070295 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( a + 2\) , \( -1\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}-1$ |
| 1225.2-d1 |
1225.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{2} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$0.726568006$ |
$14.20486200$ |
4.504365649 |
\( -112447108485120 a - 201425138913280 \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -20 a + 22\) , \( -97 a + 357\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-20a+22\right){x}-97a+357$ |
| 1225.2-d2 |
1225.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{2} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$0.242189335$ |
$42.61458602$ |
4.504365649 |
\( 184320 a - 593920 \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 10 a - 28\) , \( -30 a + 82\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(10a-28\right){x}-30a+82$ |
| 1225.2-e1 |
1225.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{8} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$5.077814817$ |
$1.279667003$ |
5.671842653 |
\( -\frac{2696721}{175} a - \frac{4569949}{175} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -129 a + 355\) , \( -368 a + 1017\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-129a+355\right){x}-368a+1017$ |
| 1225.2-e2 |
1225.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{12} \cdot 7^{12} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$15.23344445$ |
$0.426555667$ |
5.671842653 |
\( \frac{6353908769}{5359375} a - \frac{1663680002}{765625} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 1446 a - 4020\) , \( 52258 a - 145878\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1446a-4020\right){x}+52258a-145878$ |
| 1225.2-e3 |
1225.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{18} \cdot 7^{9} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$7.616722226$ |
$0.426555667$ |
5.671842653 |
\( -\frac{65314499704061021}{11962890625} a + \frac{182342243113214601}{11962890625} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( -9590 a - 17388\) , \( -456883 a - 819448\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-9590a-17388\right){x}-456883a-819448$ |
| 1225.2-e4 |
1225.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{10} \cdot 7^{7} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$2.538907408$ |
$1.279667003$ |
5.671842653 |
\( \frac{4660480434957}{4375} a + \frac{8348270709308}{4375} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 536 a - 1570\) , \( -4470 a + 12182\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(536a-1570\right){x}-4470a+12182$ |
| 1225.2-f1 |
1225.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{8} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$8.553414804$ |
$1.647291830$ |
6.149367196 |
\( -112447108485120 a - 201425138913280 \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( -2007 a - 3323\) , \( 73161 a + 125713\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2007a-3323\right){x}+73161a+125713$ |
| 1225.2-f2 |
1225.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{8} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$2.851138268$ |
$1.647291830$ |
6.149367196 |
\( 184320 a - 593920 \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( 23 a - 173\) , \( 424 a - 672\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(23a-173\right){x}+424a-672$ |
| 1225.2-g1 |
1225.2-g |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{2} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$4.119702204$ |
$0.903785405$ |
1.624993006 |
\( -112447108485120 a - 201425138913280 \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( -172 a - 303\) , \( -1945 a - 3493\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-172a-303\right){x}-1945a-3493$ |
| 1225.2-g2 |
1225.2-g |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{2} \cdot 7^{2} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1.373234068$ |
$2.711356217$ |
1.624993006 |
\( 184320 a - 593920 \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( -2 a - 3\) , \( -4 a - 8\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a-3\right){x}-4a-8$ |
| 1225.2-h1 |
1225.2-h |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{10} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 1 \) |
$2.114519451$ |
$4.809219702$ |
4.438197767 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( a + 2\) , \( 755 a + 1353\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}+755a+1353$ |
| 1225.2-h2 |
1225.2-h |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{10} \cdot 7^{6} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$7$ |
7Ns.3.1 |
$1$ |
\( 1 \) |
$6.343558355$ |
$1.603073234$ |
4.438197767 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( a + 2\) , \( 10 a - 61\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}+10a-61$ |
| 1225.2-i1 |
1225.2-i |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{8} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.810454642$ |
$4.106194335$ |
2.904815529 |
\( -\frac{2696721}{175} a - \frac{4569949}{175} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -28 a - 25\) , \( 59 a + 95\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-28a-25\right){x}+59a+95$ |
| 1225.2-i2 |
1225.2-i |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{12} \cdot 7^{12} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$2.431363926$ |
$1.368731445$ |
2.904815529 |
\( \frac{6353908769}{5359375} a - \frac{1663680002}{765625} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 147 a - 25\) , \( 122 a + 2335\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(147a-25\right){x}+122a+2335$ |
| 1225.2-i3 |
1225.2-i |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{18} \cdot 7^{9} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1.215681963$ |
$1.368731445$ |
2.904815529 |
\( -\frac{65314499704061021}{11962890625} a + \frac{182342243113214601}{11962890625} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 637 a - 3700\) , \( -24476 a + 83920\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(637a-3700\right){x}-24476a+83920$ |
| 1225.2-i4 |
1225.2-i |
$4$ |
$6$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{10} \cdot 7^{7} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$0.405227321$ |
$4.106194335$ |
2.904815529 |
\( \frac{4660480434957}{4375} a + \frac{8348270709308}{4375} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -343 a - 725\) , \( 5288 a + 9265\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-343a-725\right){x}+5288a+9265$ |
| 1225.2-j1 |
1225.2-j |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{8} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$11.76736065$ |
$0.222671048$ |
3.430713264 |
\( -112447108485120 a - 201425138913280 \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -1435 a + 3617\) , \( -135298 a + 374573\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(-1435a+3617\right){x}-135298a+374573$ |
| 1225.2-j2 |
1225.2-j |
$2$ |
$3$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1225.2 |
\( 5^{2} \cdot 7^{2} \) |
\( 5^{8} \cdot 7^{8} \) |
$2.42260$ |
$(-a), (a+3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$3.922453551$ |
$2.004039437$ |
3.430713264 |
\( 184320 a - 593920 \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( -548 a - 990\) , \( -10244 a - 18345\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(-548a-990\right){x}-10244a-18345$ |
*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.
