| 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 |
| 2.1-a1 |
2.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
2.1 |
\( 2 \) |
\( 2^{24} \) |
$1.60459$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$5.773194903$ |
2.294035035 |
\( \frac{2146689}{64} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -15 a + 43\) , \( 34 a + 35\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-15a+43\right){x}+34a+35$ |
| 2.1-a2 |
2.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
2.1 |
\( 2 \) |
\( 2^{18} \) |
$1.60459$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$2.886597451$ |
2.294035035 |
\( \frac{8602523649}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -15 a + 203\) , \( 2 a - 765\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-15a+203\right){x}+2a-765$ |
| 2.1-b1 |
2.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
2.1 |
\( 2 \) |
\( 2^{24} \) |
$1.60459$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$5.773194903$ |
2.294035035 |
\( \frac{2146689}{64} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -16 a + 71\) , \( 37 a + 49\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-16a+71\right){x}+37a+49$ |
| 2.1-b2 |
2.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
2.1 |
\( 2 \) |
\( 2^{18} \) |
$1.60459$ |
$(2,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$2.886597451$ |
2.294035035 |
\( \frac{8602523649}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -16 a + 231\) , \( 229 a - 751\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-16a+231\right){x}+229a-751$ |
| 2.1-c1 |
2.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
2.1 |
\( 2 \) |
\( 2^{12} \cdot 11^{12} \) |
$1.60459$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$2.422755826$ |
$5.773194903$ |
1.852628916 |
\( \frac{2146689}{64} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -43 a + 19\) , \( 198 a - 1688\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2+\left(-43a+19\right){x}+198a-1688$ |
| 2.1-c2 |
2.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
2.1 |
\( 2 \) |
\( 2^{6} \cdot 11^{12} \) |
$1.60459$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$4$ |
\( 2 \) |
$1.211377913$ |
$2.886597451$ |
1.852628916 |
\( \frac{8602523649}{8} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -683 a + 299\) , \( 12342 a - 109488\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2+\left(-683a+299\right){x}+12342a-109488$ |
| 2.1-d1 |
2.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
2.1 |
\( 2 \) |
\( 2^{12} \cdot 11^{12} \) |
$1.60459$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$2.422755826$ |
$5.773194903$ |
1.852628916 |
\( \frac{2146689}{64} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -43 a + 115\) , \( 17 a + 1501\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-43a+115\right){x}+17a+1501$ |
| 2.1-d2 |
2.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
2.1 |
\( 2 \) |
\( 2^{6} \cdot 11^{12} \) |
$1.60459$ |
$(2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Nn |
$4$ |
\( 2 \) |
$1.211377913$ |
$2.886597451$ |
1.852628916 |
\( \frac{8602523649}{8} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -683 a + 395\) , \( -8927 a + 107901\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-683a+395\right){x}-8927a+107901$ |
| 9.1-a1 |
9.1-a |
$2$ |
$19$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \cdot 11^{12} \) |
$2.33704$ |
$(3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19B.18.6 |
$1$ |
\( 2 \) |
$1$ |
$5.069340169$ |
1.342901016 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -96 a + 42\) , \( -642 a + 5771\bigr] \) |
${y}^2+{y}={x}^3+\left(-96a+42\right){x}-642a+5771$ |
| 9.1-a2 |
9.1-a |
$2$ |
$19$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$2.33704$ |
$(3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19B.18.6 |
$1$ |
\( 2 \) |
$1$ |
$5.069340169$ |
1.342901016 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 24\) , \( -6 a\bigr] \) |
${y}^2={x}^3+24{x}-6a$ |
| 9.1-b1 |
9.1-b |
$2$ |
$19$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
9.1 |
\( 3^{2} \) |
\( 3^{6} \cdot 11^{12} \) |
$2.33704$ |
$(3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19B.18.6 |
$1$ |
\( 2 \) |
$1$ |
$5.069340169$ |
1.342901016 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -96 a + 42\) , \( 642 a - 5757\bigr] \) |
${y}^2+a{y}={x}^3+\left(-96a+42\right){x}+642a-5757$ |
| 9.1-b2 |
9.1-b |
$2$ |
$19$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
9.1 |
\( 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$2.33704$ |
$(3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-19$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$19$ |
19B.18.6 |
$1$ |
\( 2 \) |
$1$ |
$5.069340169$ |
1.342901016 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 24\) , \( 6 a\bigr] \) |
${y}^2={x}^3+24{x}+6a$ |
| 18.1-a1 |
18.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{34} \cdot 11^{12} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$0.373800225$ |
1.584353578 |
\( \frac{248028267187}{76527504} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -6283 a + 2730\) , \( -231370 a + 1965754\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(-6283a+2730\right){x}-231370a+1965754$ |
| 18.1-a2 |
18.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{20} \cdot 11^{12} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.747600450$ |
1.584353578 |
\( \frac{14580432307}{559872} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -2443 a + 1050\) , \( 79814 a - 761846\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(-2443a+1050\right){x}+79814a-761846$ |
| 18.1-b1 |
18.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{34} \cdot 11^{12} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$0.373800225$ |
1.584353578 |
\( \frac{248028267187}{76527504} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -6293 a + 2826\) , \( 262807 a - 1979482\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(-6293a+2826\right){x}+262807a-1979482$ |
| 18.1-b2 |
18.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{16} \cdot 3^{20} \cdot 11^{12} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.747600450$ |
1.584353578 |
\( \frac{14580432307}{559872} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -2453 a + 1146\) , \( -67577 a + 756518\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^3+a{x}^2+\left(-2453a+1146\right){x}-67577a+756518$ |
| 18.1-c1 |
18.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{34} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$8.320003836$ |
$0.373800225$ |
6.590913924 |
\( \frac{248028267187}{76527504} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -6 a + 1669\) , \( 2442 a - 7941\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-6a+1669\right){x}+2442a-7941$ |
| 18.1-c2 |
18.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{20} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$4.160001918$ |
$0.747600450$ |
6.590913924 |
\( \frac{14580432307}{559872} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -6 a + 709\) , \( -630 a - 3141\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-6a+709\right){x}-630a-3141$ |
| 18.1-d1 |
18.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{34} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$8.320003836$ |
$0.373800225$ |
6.590913924 |
\( \frac{248028267187}{76527504} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 4 a + 1669\) , \( -2443 a - 7941\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(4a+1669\right){x}-2443a-7941$ |
| 18.1-d2 |
18.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{20} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$4.160001918$ |
$0.747600450$ |
6.590913924 |
\( \frac{14580432307}{559872} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 4 a + 709\) , \( 629 a - 3141\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(4a+709\right){x}+629a-3141$ |
| 18.1-e1 |
18.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{6} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$5.730072601$ |
1.517933313 |
\( -\frac{27}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -16 a + 61\) , \( 31 a + 99\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-16a+61\right){x}+31a+99$ |
| 18.1-f1 |
18.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{18} \cdot 3^{6} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \) |
$1$ |
$5.730072601$ |
1.517933313 |
\( -\frac{27}{8} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -15 a + 33\) , \( 30 a + 85\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-15a+33\right){x}+30a+85$ |
| 18.1-g1 |
18.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \cdot 11^{12} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$0.729810716$ |
$5.730072601$ |
6.646823996 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -3 a + 69\) , \( -66 a + 608\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-3a+69\right){x}-66a+608$ |
| 18.1-h1 |
18.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
18.1 |
\( 2 \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \cdot 11^{12} \) |
$2.77923$ |
$(2,a+1), (3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Nn |
$1$ |
\( 2 \cdot 3 \) |
$0.729810716$ |
$5.730072601$ |
6.646823996 |
\( -\frac{27}{8} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3 a + 1\) , \( 81 a - 722\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(-3a+1\right){x}+81a-722$ |
| 19.1-a1 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$11.02710220$ |
$0.935309008$ |
5.464357194 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -6924\) , \( -221746\bigr] \) |
${y}^2+a{y}={x}^3-6924{x}-221746$ |
| 19.1-a2 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{6} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$3.675700734$ |
$2.805927025$ |
5.464357194 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -84\) , \( -301\bigr] \) |
${y}^2+a{y}={x}^3-84{x}-301$ |
| 19.1-a3 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1.225233578$ |
$8.417781075$ |
5.464357194 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 6\) , \( 14\bigr] \) |
${y}^2+a{y}={x}^3+6{x}+14$ |
| 19.1-b1 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$33.64532080$ |
$0.935309008$ |
3.705013890 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -6924\) , \( 221760\bigr] \) |
${y}^2+{y}={x}^3-6924{x}+221760$ |
| 19.1-b2 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{6} \) |
$2.81705$ |
$(19,a)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.738368977$ |
$2.805927025$ |
3.705013890 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -84\) , \( 315\bigr] \) |
${y}^2+{y}={x}^3-84{x}+315$ |
| 19.1-b3 |
19.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 3^{12} \cdot 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \) |
$0.415374330$ |
$8.417781075$ |
3.705013890 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 6\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3+6{x}$ |
| 19.1-c1 |
19.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$9.202414482$ |
$0.935309008$ |
4.560153597 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -769\) , \( 8484\bigr] \) |
${y}^2+a{y}={x}^3-{x}^2-769{x}+8484$ |
| 19.1-c2 |
19.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$3.067471494$ |
$2.805927025$ |
4.560153597 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -9\) , \( 29\bigr] \) |
${y}^2+a{y}={x}^3-{x}^2-9{x}+29$ |
| 19.1-c3 |
19.1-c |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.81705$ |
$(19,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$1.022490498$ |
$8.417781075$ |
4.560153597 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( 1\) , \( 14\bigr] \) |
${y}^2+a{y}={x}^3-{x}^2+{x}+14$ |
| 19.1-d1 |
19.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.81705$ |
$(19,a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$4$ |
\( 2 \) |
$1$ |
$0.935309008$ |
0.991077636 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -769\) , \( -8470\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-769{x}-8470$ |
| 19.1-d2 |
19.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$2.81705$ |
$(19,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$4$ |
\( 2 \cdot 3 \) |
$1$ |
$2.805927025$ |
0.991077636 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -9\) , \( -15\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-9{x}-15$ |
| 19.1-d3 |
19.1-d |
$3$ |
$9$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$2.81705$ |
$(19,a)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$4$ |
\( 2 \) |
$1$ |
$8.417781075$ |
0.991077636 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3+{x}^2+{x}$ |
| 24.1-a1 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.562595083$ |
$1.817673508$ |
6.019284716 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -59 a + 116\) , \( -2672 a - 19486\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-59a+116\right){x}-2672a-19486$ |
| 24.1-a2 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.125190167$ |
$7.270694035$ |
6.019284716 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( 7\bigr] \) |
${y}^2={x}^3+6{x}+7$ |
| 24.1-a3 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$6.250380335$ |
$7.270694035$ |
6.019284716 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 21 a + 81\) , \( -47 a + 412\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(21a+81\right){x}-47a+412$ |
| 24.1-a4 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$3.125190167$ |
$3.635347017$ |
6.019284716 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 101 a + 46\) , \( -1202 a - 3658\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(101a+46\right){x}-1202a-3658$ |
| 24.1-a5 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$12.50076067$ |
$3.635347017$ |
6.019284716 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 261 a - 24\) , \( 2158 a + 24154\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(261a-24\right){x}+2158a+24154$ |
| 24.1-a6 |
24.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$6.250380335$ |
$1.817673508$ |
6.019284716 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 1541 a - 584\) , \( -56012 a - 292630\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(1541a-584\right){x}-56012a-292630$ |
| 24.1-b1 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{16} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$1.817673508$ |
0.963026950 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -67 a + 79\) , \( 3234 a + 19113\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(-67a+79\right){x}+3234a+19113$ |
| 24.1-b2 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{14} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$7.270694035$ |
0.963026950 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( -7\bigr] \) |
${y}^2={x}^3+6{x}-7$ |
| 24.1-b3 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$7.270694035$ |
0.963026950 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 13 a + 44\) , \( -111 a - 470\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(13a+44\right){x}-111a-470$ |
| 24.1-b4 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$3.635347017$ |
0.963026950 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 93 a + 9\) , \( 324 a + 3915\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(93a+9\right){x}+324a+3915$ |
| 24.1-b5 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.635347017$ |
0.963026950 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 253 a - 61\) , \( -4476 a - 23267\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(253a-61\right){x}-4476a-23267$ |
| 24.1-b6 |
24.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{10} \cdot 3^{4} \cdot 11^{12} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$1$ |
$1.817673508$ |
0.963026950 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 1533 a - 621\) , \( 42174 a + 298557\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(1533a-621\right){x}+42174a+298557$ |
| 24.1-c1 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{22} \cdot 3^{16} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$16$ |
\( 2^{5} \) |
$1$ |
$1.817673508$ |
1.926053901 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 16\) , \( 180\bigr] \) |
${y}^2={x}^3+{x}^2+16{x}+180$ |
| 24.1-c2 |
24.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-57}) \) |
$2$ |
$[0, 1]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{2} \) |
$2.98647$ |
$(2,a+1), (3,a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$7.270694035$ |
1.926053901 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^3+{x}^2+{x}$ |
*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.
