| 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 |
| 17.1-a1 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 17^{8} \) |
$1.29579$ |
$(17,a+8)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.123938699$ |
0.297410905 |
\( -\frac{35937}{83521} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( -14\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-{x}-14$ |
| 17.1-a2 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$1.29579$ |
$(17,a+8)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.495754796$ |
0.297410905 |
\( \frac{35937}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-{x}$ |
| 17.1-a3 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$1.29579$ |
$(17,a+8)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.247877398$ |
0.297410905 |
\( \frac{20346417}{289} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -6\) , \( -4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-6{x}-4$ |
| 17.1-a4 |
17.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$1.29579$ |
$(17,a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.123938699$ |
0.297410905 |
\( \frac{82483294977}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -91\) , \( -310\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2-91{x}-310$ |
| 17.1-b1 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{8} \) |
$1.29579$ |
$(17,a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.035256177$ |
$2.123938699$ |
1.805436579 |
\( -\frac{35937}{83521} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -6\) , \( 377\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-6{x}+377$ |
| 17.1-b2 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{2} \) |
$1.29579$ |
$(17,a+8)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.035256177$ |
$8.495754796$ |
1.805436579 |
\( \frac{35937}{17} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -6\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-6{x}-1$ |
| 17.1-b3 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{4} \) |
$1.29579$ |
$(17,a+8)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$6.070512355$ |
$4.247877398$ |
1.805436579 |
\( \frac{20346417}{289} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -51\) , \( 152\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-51{x}+152$ |
| 17.1-b4 |
17.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
17.1 |
\( 17 \) |
\( 3^{12} \cdot 17^{2} \) |
$1.29579$ |
$(17,a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.035256177$ |
$2.123938699$ |
1.805436579 |
\( \frac{82483294977}{17} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -816\) , \( 9179\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-816{x}+9179$ |
| 45.1-a1 |
45.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{13} \cdot 5^{7} \) |
$1.65282$ |
$(3,a+1), (5,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$1.524104971$ |
1.707339070 |
\( -\frac{7882339931}{6328125} a + \frac{34505684782}{6328125} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 3 a - 12\) , \( 4 a\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(3a-12\right){x}+4a$ |
| 45.1-b1 |
45.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{25} \cdot 5^{7} \) |
$1.65282$ |
$(3,a+1), (5,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.247061616$ |
$1.524104971$ |
2.952725655 |
\( -\frac{7882339931}{6328125} a + \frac{34505684782}{6328125} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 43 a - 219\) , \( -327 a + 1107\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(43a-219\right){x}-327a+1107$ |
| 45.2-a1 |
45.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{13} \cdot 5^{7} \) |
$1.65282$ |
$(3,a+1), (5,a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$1.524104971$ |
1.707339070 |
\( \frac{7882339931}{6328125} a + \frac{26623344851}{6328125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -5 a - 8\) , \( -5 a + 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(-5a-8\right){x}-5a+5$ |
| 45.2-b1 |
45.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
45.2 |
\( 3^{2} \cdot 5 \) |
\( 3^{25} \cdot 5^{7} \) |
$1.65282$ |
$(3,a+1), (5,a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.247061616$ |
$1.524104971$ |
2.952725655 |
\( \frac{7882339931}{6328125} a + \frac{26623344851}{6328125} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -44 a - 176\) , \( 327 a + 780\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(-44a-176\right){x}+327a+780$ |
| 51.1-a1 |
51.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{14} \cdot 17^{6} \) |
$1.70536$ |
$(3,a+1), (17,a+8)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.689501217$ |
$1.679417732$ |
1.729567068 |
\( -\frac{23100424192}{14739} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -534\) , \( 4752\bigr] \) |
${y}^2+{y}={x}^3-534{x}+4752$ |
| 51.1-a2 |
51.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{18} \cdot 17^{2} \) |
$1.70536$ |
$(3,a+1), (17,a+8)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$0.076611246$ |
$5.038253197$ |
1.729567068 |
\( \frac{32768}{459} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 6\) , \( 27\bigr] \) |
${y}^2+{y}={x}^3+6{x}+27$ |
| 51.1-b1 |
51.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{2} \cdot 17^{6} \) |
$1.70536$ |
$(3,a+1), (17,a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1.679417732$ |
1.881324162 |
\( -\frac{23100424192}{14739} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -59\) , \( -196\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-59{x}-196$ |
| 51.1-b2 |
51.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
51.1 |
\( 3 \cdot 17 \) |
\( 3^{6} \cdot 17^{2} \) |
$1.70536$ |
$(3,a+1), (17,a+8)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$5.038253197$ |
1.881324162 |
\( \frac{32768}{459} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 1\) , \( -1\bigr] \) |
${y}^2+{y}={x}^3+{x}^2+{x}-1$ |
| 65.2-a1 |
65.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.2 |
\( 5 \cdot 13 \) |
\( 3^{12} \cdot 5^{8} \cdot 13^{2} \) |
$1.81197$ |
$(5,a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.670503523$ |
1.871338251 |
\( -\frac{45308332052343}{66015625} a - \frac{17464922443908}{5078125} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -48 a - 576\) , \( -771 a - 4433\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-48a-576\right){x}-771a-4433$ |
| 65.2-a2 |
65.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.2 |
\( 5 \cdot 13 \) |
\( 3^{12} \cdot 5^{2} \cdot 13^{8} \) |
$1.81197$ |
$(5,a+1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.670503523$ |
1.871338251 |
\( -\frac{67798096507449}{20393268025} a - \frac{15058954662444}{1568712925} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 42 a - 216\) , \( -465 a + 1057\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(42a-216\right){x}-465a+1057$ |
| 65.2-a3 |
65.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.2 |
\( 5 \cdot 13 \) |
\( 3^{12} \cdot 5^{4} \cdot 13^{4} \) |
$1.81197$ |
$(5,a+1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$3.341007047$ |
1.871338251 |
\( \frac{9747065097}{17850625} a - \frac{221169393}{1373125} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -3 a - 36\) , \( -24 a - 32\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-3a-36\right){x}-24a-32$ |
| 65.2-a4 |
65.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.2 |
\( 5 \cdot 13 \) |
\( 3^{12} \cdot 5^{2} \cdot 13^{2} \) |
$1.81197$ |
$(5,a+1), (a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$6.682014095$ |
1.871338251 |
\( -\frac{2508489}{4225} a + \frac{148716}{325} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -3 a + 9\) , \( 3 a - 14\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-3a+9\right){x}+3a-14$ |
| 65.2-b1 |
65.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.2 |
\( 5 \cdot 13 \) |
\( 5^{8} \cdot 13^{2} \) |
$1.81197$ |
$(5,a+1), (a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$3.072952641$ |
$1.670503523$ |
2.875266912 |
\( -\frac{45308332052343}{66015625} a - \frac{17464922443908}{5078125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -6 a - 53\) , \( 36 a + 272\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-6a-53\right){x}+36a+272$ |
| 65.2-b2 |
65.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.2 |
\( 5 \cdot 13 \) |
\( 5^{2} \cdot 13^{8} \) |
$1.81197$ |
$(5,a+1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.768238160$ |
$1.670503523$ |
2.875266912 |
\( -\frac{67798096507449}{20393268025} a - \frac{15058954662444}{1568712925} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 4 a - 13\) , \( 8 a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(4a-13\right){x}+8a+2$ |
| 65.2-b3 |
65.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.2 |
\( 5 \cdot 13 \) |
\( 5^{4} \cdot 13^{4} \) |
$1.81197$ |
$(5,a+1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.536476320$ |
$3.341007047$ |
2.875266912 |
\( \frac{9747065097}{17850625} a - \frac{221169393}{1373125} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -a + 7\) , \( 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-a+7\right){x}+9$ |
| 65.2-b4 |
65.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.2 |
\( 5 \cdot 13 \) |
\( 5^{2} \cdot 13^{2} \) |
$1.81197$ |
$(5,a+1), (a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.072952641$ |
$6.682014095$ |
2.875266912 |
\( -\frac{2508489}{4225} a + \frac{148716}{325} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -a + 12\) , \( -a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a-1\right){x}^2+\left(-a+12\right){x}-a$ |
| 65.3-a1 |
65.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 5^{8} \cdot 13^{2} \) |
$1.81197$ |
$(5,a+3), (a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$3.072952641$ |
$1.670503523$ |
2.875266912 |
\( \frac{45308332052343}{66015625} a - \frac{272352323823147}{66015625} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 4 a - 57\) , \( -37 a + 309\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(4a-57\right){x}-37a+309$ |
| 65.3-a2 |
65.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 5^{2} \cdot 13^{8} \) |
$1.81197$ |
$(5,a+3), (a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.768238160$ |
$1.670503523$ |
2.875266912 |
\( \frac{67798096507449}{20393268025} a - \frac{263564507119221}{20393268025} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -6 a - 7\) , \( -9 a + 11\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-6a-7\right){x}-9a+11$ |
| 65.3-a3 |
65.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 5^{2} \cdot 13^{2} \) |
$1.81197$ |
$(5,a+3), (a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.072952641$ |
$6.682014095$ |
2.875266912 |
\( \frac{2508489}{4225} a - \frac{575181}{4225} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -a + 13\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-a+13\right){x}$ |
| 65.3-a4 |
65.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 5^{4} \cdot 13^{4} \) |
$1.81197$ |
$(5,a+3), (a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.536476320$ |
$3.341007047$ |
2.875266912 |
\( -\frac{9747065097}{17850625} a + \frac{6871862988}{17850625} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -a + 8\) , \( -a + 10\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-a+8\right){x}-a+10$ |
| 65.3-b1 |
65.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 3^{12} \cdot 5^{8} \cdot 13^{2} \) |
$1.81197$ |
$(5,a+3), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.670503523$ |
1.871338251 |
\( \frac{45308332052343}{66015625} a - \frac{272352323823147}{66015625} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 48 a - 624\) , \( 771 a - 5204\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(48a-624\right){x}+771a-5204$ |
| 65.3-b2 |
65.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 3^{12} \cdot 5^{2} \cdot 13^{8} \) |
$1.81197$ |
$(5,a+3), (a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.670503523$ |
1.871338251 |
\( \frac{67798096507449}{20393268025} a - \frac{263564507119221}{20393268025} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -42 a - 174\) , \( 465 a + 592\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(-42a-174\right){x}+465a+592$ |
| 65.3-b3 |
65.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 3^{12} \cdot 5^{2} \cdot 13^{2} \) |
$1.81197$ |
$(5,a+3), (a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$6.682014095$ |
1.871338251 |
\( \frac{2508489}{4225} a - \frac{575181}{4225} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 3 a + 6\) , \( -3 a - 11\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(3a+6\right){x}-3a-11$ |
| 65.3-b4 |
65.3-b |
$4$ |
$4$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
65.3 |
\( 5 \cdot 13 \) |
\( 3^{12} \cdot 5^{4} \cdot 13^{4} \) |
$1.81197$ |
$(5,a+3), (a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$3.341007047$ |
1.871338251 |
\( -\frac{9747065097}{17850625} a + \frac{6871862988}{17850625} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 3 a - 39\) , \( 24 a - 56\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(3a-39\right){x}+24a-56$ |
| 67.1-a1 |
67.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
67.1 |
\( 67 \) |
\( 3^{12} \cdot 67 \) |
$1.82575$ |
$(-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$9.070165672$ |
2.540154469 |
\( -\frac{33238}{67} a - \frac{291861}{67} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -a + 2\) , \( 3 a + 5\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(-a+2\right){x}+3a+5$ |
| 67.1-b1 |
67.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
67.1 |
\( 67 \) |
\( 67 \) |
$1.82575$ |
$(-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$9.070165672$ |
2.540154469 |
\( -\frac{33238}{67} a - \frac{291861}{67} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -2 a + 1\) , \( -a + 8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-2a+1\right){x}-a+8$ |
| 67.2-a1 |
67.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
67.2 |
\( 67 \) |
\( 3^{12} \cdot 67 \) |
$1.82575$ |
$(2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$9.070165672$ |
2.540154469 |
\( \frac{33238}{67} a - \frac{325099}{67} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -a + 2\) , \( -4 a + 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^3+\left(-a+2\right){x}-4a+9$ |
| 67.2-b1 |
67.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
67.2 |
\( 67 \) |
\( 67 \) |
$1.82575$ |
$(2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$9.070165672$ |
2.540154469 |
\( \frac{33238}{67} a - \frac{325099}{67} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -4 a - 9\) , \( -a + 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(a-1\right){x}^2+\left(-4a-9\right){x}-a+12$ |
| 68.1-a1 |
68.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{12} \cdot 17^{2} \) |
$1.83253$ |
$(17,a+8), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.190351719$ |
1.564710948 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^3-3{x}+1$ |
| 68.1-a2 |
68.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 17^{12} \) |
$1.83253$ |
$(17,a+8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$16$ |
\( 2 \) |
$1$ |
$0.698391953$ |
1.564710948 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \) |
${y}^2+{x}{y}={x}^3-113{x}-329$ |
| 68.1-a3 |
68.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{6} \cdot 17^{4} \) |
$1.83253$ |
$(17,a+8), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$16$ |
\( 2 \cdot 3 \) |
$1$ |
$2.095175859$ |
1.564710948 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \) |
${y}^2+{x}{y}={x}^3-43{x}+105$ |
| 68.1-a4 |
68.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{4} \cdot 17^{6} \) |
$1.83253$ |
$(17,a+8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2^{2} \) |
$1$ |
$1.396783906$ |
1.564710948 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \) |
${y}^2+{x}{y}={x}^3-103{x}-411$ |
| 68.1-b1 |
68.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{12} \cdot 3^{12} \cdot 17^{2} \) |
$1.83253$ |
$(17,a+8), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.562629009$ |
$4.190351719$ |
3.961582973 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -27\) , \( -27\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-27{x}-27$ |
| 68.1-b2 |
68.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{2} \cdot 3^{12} \cdot 17^{12} \) |
$1.83253$ |
$(17,a+8), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \cdot 3 \) |
$3.375774058$ |
$0.698391953$ |
3.961582973 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -1017\) , \( 8883\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-1017{x}+8883$ |
| 68.1-b3 |
68.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{6} \cdot 3^{12} \cdot 17^{4} \) |
$1.83253$ |
$(17,a+8), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.125258019$ |
$2.095175859$ |
3.961582973 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -387\) , \( -2835\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-387{x}-2835$ |
| 68.1-b4 |
68.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
68.1 |
\( 2^{2} \cdot 17 \) |
\( 2^{4} \cdot 3^{12} \cdot 17^{6} \) |
$1.83253$ |
$(17,a+8), (2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \cdot 3 \) |
$1.687887029$ |
$1.396783906$ |
3.961582973 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( -1\) , \( 0\) , \( -927\) , \( 11097\bigr] \) |
${y}^2+{x}{y}={x}^3-{x}^2-927{x}+11097$ |
| 75.1-a1 |
75.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{13} \cdot 5^{15} \) |
$1.87797$ |
$(3,a+1), (5,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1.339218087$ |
$1.468487376$ |
2.203060494 |
\( -\frac{4580785442}{5859375} a - \frac{97087414301}{5859375} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -89 a + 96\) , \( -282 a + 1822\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-89a+96\right){x}-282a+1822$ |
| 75.1-b1 |
75.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{22} \cdot 5^{6} \) |
$1.87797$ |
$(3,a+1), (5,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cn, 5B |
$1$ |
\( 2 \) |
$1.815389237$ |
$1.026483166$ |
2.087500014 |
\( -\frac{13549359104}{243} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 447 a - 1788\) , \( 10587 a - 20292\bigr] \) |
${y}^2+{y}={x}^3+\left(447a-1788\right){x}+10587a-20292$ |
| 75.1-b2 |
75.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{14} \cdot 5^{6} \) |
$1.87797$ |
$(3,a+1), (5,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 5$ |
2Cn, 5B |
$1$ |
\( 2 \) |
$0.363077847$ |
$5.132415832$ |
2.087500014 |
\( \frac{4096}{3} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -3 a + 12\) , \( 3 a - 6\bigr] \) |
${y}^2+{y}={x}^3+\left(-3a+12\right){x}+3a-6$ |
| 75.1-c1 |
75.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{16} \cdot 5^{6} \) |
$1.87797$ |
$(3,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$4.114216953$ |
$1.430770648$ |
3.297099965 |
\( \frac{5359375}{6561} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -4 a + 16\) , \( 12 a - 23\bigr] \) |
${y}^2+a{x}{y}={x}^3+{x}^2+\left(-4a+16\right){x}+12a-23$ |
| 75.1-c2 |
75.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3^{8} \cdot 5^{6} \) |
$1.87797$ |
$(3,a+1), (5,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.057108476$ |
$2.861541296$ |
3.297099965 |
\( \frac{274625}{81} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( a - 4\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^3+{x}^2+\left(a-4\right){x}$ |
| 75.1-d1 |
75.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{-51}) \) |
$2$ |
$[0, 1]$ |
75.1 |
\( 3 \cdot 5^{2} \) |
\( 3 \cdot 5^{15} \) |
$1.87797$ |
$(3,a+1), (5,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$1.468487376$ |
1.645034901 |
\( -\frac{4580785442}{5859375} a - \frac{97087414301}{5859375} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -11 a + 14\) , \( 24 a - 83\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-11a+14\right){x}+24a-83$ |
*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.
