| 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 |
| 45.1-a1 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{32} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$6.492249124$ |
0.774245886 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -35 a^{3} + 174 a^{2} - 140 a - 32\) , \( -328 a^{3} + 1241 a^{2} - 884 a - 245\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(-35a^{3}+174a^{2}-140a-32\right){x}-328a^{3}+1241a^{2}-884a-245$ |
| 45.1-a2 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$103.8759859$ |
0.774245886 |
\( -\frac{1}{15} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - 5 a^{2} + 5 a\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(-a^{2}+3\right){x}+a^{3}-5a^{2}+5a$ |
| 45.1-a3 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$103.8759859$ |
0.774245886 |
\( \frac{13997521}{225} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 5 a^{3} - 26 a^{2} + 20 a + 8\) , \( 32 a^{3} - 129 a^{2} + 96 a + 25\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(5a^{3}-26a^{2}+20a+8\right){x}+32a^{3}-129a^{2}+96a+25$ |
| 45.1-a4 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{16} \cdot 5^{16} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$103.8759859$ |
0.774245886 |
\( \frac{111284641}{50625} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 10 a^{3} - 51 a^{2} + 40 a + 13\) , \( -49 a^{3} + 125 a^{2} - 65 a - 20\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(10a^{3}-51a^{2}+40a+13\right){x}-49a^{3}+125a^{2}-65a-20$ |
| 45.1-a5 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$6.492249124$ |
0.774245886 |
\( \frac{56667352321}{15} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 80 a^{3} - 401 a^{2} + 320 a + 83\) , \( 2177 a^{3} - 7659 a^{2} + 5241 a + 1450\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(80a^{3}-401a^{2}+320a+83\right){x}+2177a^{3}-7659a^{2}+5241a+1450$ |
| 45.1-a6 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{64} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$6.492249124$ |
0.774245886 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 110 a^{3} - 551 a^{2} + 440 a + 113\) , \( -6709 a^{3} + 22215 a^{2} - 14625 a - 4070\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(110a^{3}-551a^{2}+440a+113\right){x}-6709a^{3}+22215a^{2}-14625a-4070$ |
| 45.1-a7 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{32} \cdot 5^{8} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$103.8759859$ |
0.774245886 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 135 a^{3} - 676 a^{2} + 540 a + 138\) , \( -4874 a^{3} + 15925 a^{2} - 10390 a - 2895\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(135a^{3}-676a^{2}+540a+138\right){x}-4874a^{3}+15925a^{2}-10390a-2895$ |
| 45.1-a8 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$103.8759859$ |
0.774245886 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 2160 a^{3} - 10801 a^{2} + 8640 a + 2163\) , \( -309839 a^{3} + 1037335 a^{2} - 687955 a - 191220\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(2160a^{3}-10801a^{2}+8640a+2163\right){x}-309839a^{3}+1037335a^{2}-687955a-191220$ |
| 45.1-a9 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$25.96899649$ |
0.774245886 |
\( \frac{152409672113485069453847362}{45} a^{3} - \frac{152409672113485069453847362}{15} a + \frac{246604029693845863366701161}{45} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -17865 a^{3} - 26596 a^{2} + 58950 a + 11118\) , \( 1894864 a^{3} + 2845867 a^{2} - 6185020 a - 1357809\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(-17865a^{3}-26596a^{2}+58950a+11118\right){x}+1894864a^{3}+2845867a^{2}-6185020a-1357809$ |
| 45.1-a10 |
45.1-a |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$25.96899649$ |
0.774245886 |
\( -\frac{152409672113485069453847362}{45} a^{3} + \frac{152409672113485069453847362}{15} a + \frac{94194357580360793912853799}{45} \) |
\( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 54585 a^{3} - 157006 a^{2} + 87930 a + 25608\) , \( -21999902 a^{3} + 64866343 a^{2} - 38895250 a - 11166531\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(54585a^{3}-157006a^{2}+87930a+25608\right){x}-21999902a^{3}+64866343a^{2}-38895250a-11166531$ |
| 45.1-b1 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1024$ |
\( 2^{3} \) |
$1$ |
$0.015032122$ |
0.917854808 |
\( \frac{152409672113485069453847362}{45} a^{3} - \frac{152409672113485069453847362}{15} a + \frac{246604029693845863366701161}{45} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -14490 a^{3} + 43470 a - 25605\) , \( -1094688 a^{3} + 3284064 a - 1810794\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-14490a^{3}+43470a-25605\right){x}-1094688a^{3}+3284064a-1810794$ |
| 45.1-b2 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1024$ |
\( 2^{3} \) |
$1$ |
$0.015032122$ |
0.917854808 |
\( -\frac{152409672113485069453847362}{45} a^{3} + \frac{152409672113485069453847362}{15} a + \frac{94194357580360793912853799}{45} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 14490 a^{3} - 43470 a - 11115\) , \( 1094688 a^{3} - 3284064 a - 716106\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(14490a^{3}-43470a-11115\right){x}+1094688a^{3}-3284064a-716106$ |
| 45.1-b3 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$64$ |
\( 2^{5} \) |
$1$ |
$0.240513954$ |
0.917854808 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-2160{x}-39540$ |
| 45.1-b4 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{32} \cdot 5^{8} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{7} \) |
$1$ |
$3.848223270$ |
0.917854808 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-135{x}-660$ |
| 45.1-b5 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{64} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{7} \) |
$1$ |
$0.240513954$ |
0.917854808 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-110{x}-880$ |
| 45.1-b6 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$985.1451572$ |
0.917854808 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$ |
| 45.1-b7 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{16} \cdot 5^{16} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$1$ |
$61.57157232$ |
0.917854808 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-10{x}-10$ |
| 45.1-b8 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/2\Z\oplus\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$985.1451572$ |
0.917854808 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-5{x}+2$ |
| 45.1-b9 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$985.1451572$ |
0.917854808 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}$ |
| 45.1-b10 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{32} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{7} \) |
$1$ |
$3.848223270$ |
0.917854808 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+35{x}-28$ |
*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.
