| 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 |
| 4.1-a1 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$0.98700$ |
$(2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.025617488$ |
$26.92797896$ |
0.706586581 |
\( -\frac{38198355}{4} a - 32517508 \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a + 3\) , \( -2 a + 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3a+3\right){x}-2a+4$ |
| 4.1-a2 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{12} \) |
$0.98700$ |
$(2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$0.025617488$ |
$26.92797896$ |
0.706586581 |
\( -\frac{2197}{64} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -56 a - 177\) , \( 2930 a + 9990\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-56a-177\right){x}+2930a+9990$ |
| 4.1-a3 |
4.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$0.98700$ |
$(2)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.025617488$ |
$26.92797896$ |
0.706586581 |
\( \frac{38198355}{4} a - \frac{168268387}{4} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a\) , \( a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}+a+2$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.20883$ |
$(a+3), (a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.043744261$ |
$34.30762449$ |
1.537222753 |
\( \frac{18631}{3} a - \frac{234592}{9} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( a - 2\) , \( 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-2\right){x}+2$ |
| 9.1-b1 |
9.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$1.20883$ |
$(a+3), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$52.16825343$ |
1.669865100 |
\( -\frac{2197}{3} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -55 a - 182\) , \( 765 a + 2609\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-55a-182\right){x}+765a+2609$ |
| 9.1-b2 |
9.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{5} \) |
$1.20883$ |
$(a+3), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$26.08412671$ |
1.669865100 |
\( -\frac{33550495500962}{81} a + \frac{49264707079571}{27} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -1215 a - 4132\) , \( 22013 a + 74961\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1215a-4132\right){x}+22013a+74961$ |
| 9.1-b3 |
9.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.20883$ |
$(a+3), (a-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$52.16825343$ |
1.669865100 |
\( \frac{16194277}{9} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 1030 a - 4517\) , \( -34587 a + 152381\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(1030a-4517\right){x}-34587a+152381$ |
| 9.1-b4 |
9.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{5} \) |
$1.20883$ |
$(a+3), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$26.08412671$ |
1.669865100 |
\( \frac{33550495500962}{81} a + \frac{114243625737751}{81} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 1215 a - 5332\) , \( -20799 a + 91643\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(1215a-5332\right){x}-20799a+91643$ |
| 9.1-c1 |
9.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$1.20883$ |
$(a+3), (a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.043744261$ |
$34.30762449$ |
1.537222753 |
\( -\frac{18631}{3} a - \frac{178699}{9} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -2 a\) , \( 2\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}-2a{x}+2$ |
| 9.1-d1 |
9.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{12} \) |
$1.20883$ |
$(a+3), (a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 2^{2} \) |
$1$ |
$1.351524905$ |
0.692180128 |
\( -\frac{620650477}{729} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -3467 a - 11802\) , \( -219013 a - 745764\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3467a-11802\right){x}-219013a-745764$ |
| 12.1-a1 |
12.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$1.29897$ |
$(a+3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.150281765$ |
$10.82035372$ |
1.665608074 |
\( -\frac{22865}{324} a + \frac{114641}{324} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 29 a + 102\) , \( -962 a - 3279\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(29a+102\right){x}-962a-3279$ |
| 12.2-a1 |
12.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$1.29897$ |
$(a-4), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.150281765$ |
$10.82035372$ |
1.665608074 |
\( \frac{22865}{324} a + \frac{7648}{27} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -28 a + 130\) , \( 990 a - 4371\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-28a+130\right){x}+990a-4371$ |
| 15.2-a1 |
15.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{3} \cdot 5^{3} \) |
$1.37349$ |
$(a-4), (-a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$12.30111127$ |
1.574995907 |
\( -\frac{2420894}{3375} a - \frac{4801657}{1125} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 77 a - 351\) , \( -919 a + 4041\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(77a-351\right){x}-919a+4041$ |
| 15.2-b1 |
15.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5 \) |
$1.37349$ |
$(a-4), (-a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$18.91304207$ |
2.421566897 |
\( -\frac{2279}{15} a + \frac{3398}{5} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 4 a + 18\) , \( 6 a + 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4a+18\right){x}+6a+19$ |
| 15.2-c1 |
15.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{5} \cdot 5 \) |
$1.37349$ |
$(a-4), (-a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$3.173181485$ |
0.406284256 |
\( \frac{230157001561}{1215} a - \frac{337956079897}{405} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 3 a - 21\) , \( 17 a - 80\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(3a-21\right){x}+17a-80$ |
| 15.3-a1 |
15.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
15.3 |
\( 3 \cdot 5 \) |
\( 3^{3} \cdot 5^{3} \) |
$1.37349$ |
$(a+3), (a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$12.30111127$ |
1.574995907 |
\( \frac{2420894}{3375} a - \frac{3365173}{675} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -77 a - 259\) , \( 841 a + 2864\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-77a-259\right){x}+841a+2864$ |
| 15.3-b1 |
15.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
15.3 |
\( 3 \cdot 5 \) |
\( 3 \cdot 5 \) |
$1.37349$ |
$(a+3), (a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$18.91304207$ |
2.421566897 |
\( \frac{2279}{15} a + \frac{1583}{3} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 4 a + 8\) , \( 5 a + 14\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+8\right){x}+5a+14$ |
| 15.3-c1 |
15.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
15.3 |
\( 3 \cdot 5 \) |
\( 3^{5} \cdot 5 \) |
$1.37349$ |
$(a+3), (a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$3.173181485$ |
0.406284256 |
\( -\frac{230157001561}{1215} a - \frac{156742247626}{243} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -2 a - 3\) , \( -20 a - 66\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-2a-3\right){x}-20a-66$ |
| 20.1-a1 |
20.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5 \) |
$1.47591$ |
$(-a+5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.406297517$ |
$17.84692626$ |
1.856832273 |
\( -\frac{4201}{10} a - \frac{86911}{40} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -4 a\) , \( -2 a + 6\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}-4a{x}-2a+6$ |
| 20.1-a2 |
20.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{3} \) |
$1.47591$ |
$(-a+5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.135432505$ |
$17.84692626$ |
1.856832273 |
\( \frac{28571}{250} a + \frac{119482}{125} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -2 a + 12\) , \( -7 a + 38\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-2a+12\right){x}-7a+38$ |
| 20.2-a1 |
20.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5 \) |
$1.47591$ |
$(a+4), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.406297517$ |
$17.84692626$ |
1.856832273 |
\( \frac{4201}{10} a - \frac{20743}{8} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 2 a - 3\) , \( a + 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(2a-3\right){x}+a+5$ |
| 20.2-a2 |
20.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{3} \) |
$1.47591$ |
$(a+4), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.135432505$ |
$17.84692626$ |
1.856832273 |
\( -\frac{28571}{250} a + \frac{53507}{50} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 12\) , \( 6 a + 32\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+12{x}+6a+32$ |
| 25.1-a1 |
25.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{12} \) |
$1.56059$ |
$(-a+5), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$3.828387478$ |
4.411573091 |
\( \frac{2248091}{15625} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 530 a + 1810\) , \( -43920 a - 149549\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(530a+1810\right){x}-43920a-149549$ |
| 25.1-a2 |
25.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{6} \) |
$1.56059$ |
$(-a+5), (a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 3^{2} \) |
$1$ |
$15.31354991$ |
4.411573091 |
\( \frac{1295029}{125} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -445 a - 1510\) , \( -9069 a - 30877\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-445a-1510\right){x}-9069a-30877$ |
| 27.1-a1 |
27.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$1.59091$ |
$(a+3), (a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.773550333$ |
2.956909483 |
\( \frac{18631}{3} a - \frac{234592}{9} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 4\) , \( -2\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+4{x}-2$ |
| 27.1-b1 |
27.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.913279723$ |
$13.13973908$ |
1.609423347 |
\( -\frac{2197}{3} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 9 a - 33\) , \( 57 a - 261\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(9a-33\right){x}+57a-261$ |
| 27.1-b2 |
27.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{11} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.913279723$ |
$3.284934771$ |
1.609423347 |
\( -\frac{33550495500962}{81} a + \frac{49264707079571}{27} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 2699 a - 11883\) , \( 153191 a - 674835\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2699a-11883\right){x}+153191a-674835$ |
| 27.1-b3 |
27.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.956639861$ |
$13.13973908$ |
1.609423347 |
\( \frac{16194277}{9} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 169 a - 738\) , \( 2504 a - 11040\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(169a-738\right){x}+2504a-11040$ |
| 27.1-b4 |
27.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{11} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.478319930$ |
$13.13973908$ |
1.609423347 |
\( \frac{33550495500962}{81} a + \frac{114243625737751}{81} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 199 a - 873\) , \( 1625 a - 7161\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(199a-873\right){x}+1625a-7161$ |
| 27.1-c1 |
27.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{10} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.109516830$ |
$17.20222275$ |
1.929703145 |
\( -\frac{18631}{3} a - \frac{178699}{9} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -128 a - 435\) , \( 1478 a + 5032\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-128a-435\right){x}+1478a+5032$ |
| 27.1-d1 |
27.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{18} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Ns |
$1$ |
\( 2^{2} \) |
$0.479224232$ |
$3.391723688$ |
1.664885244 |
\( -\frac{620650477}{729} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 569 a - 2501\) , \( -14103 a + 62114\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(569a-2501\right){x}-14103a+62114$ |
| 27.2-a1 |
27.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{10} \) |
$1.59091$ |
$(a+3), (a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$5.773550333$ |
2.956909483 |
\( -\frac{18631}{3} a - \frac{178699}{9} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -a + 5\) , \( -a - 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-a+5\right){x}-a-1$ |
| 27.2-b1 |
27.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{8} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.913279723$ |
$13.13973908$ |
1.609423347 |
\( -\frac{2197}{3} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -8 a - 24\) , \( -66 a - 228\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-8a-24\right){x}-66a-228$ |
| 27.2-b2 |
27.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{11} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.478319930$ |
$13.13973908$ |
1.609423347 |
\( -\frac{33550495500962}{81} a + \frac{49264707079571}{27} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -198 a - 674\) , \( -1824 a - 6210\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-198a-674\right){x}-1824a-6210$ |
| 27.2-b3 |
27.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{10} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$0.956639861$ |
$13.13973908$ |
1.609423347 |
\( \frac{16194277}{9} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -168 a - 569\) , \( -2673 a - 9105\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-168a-569\right){x}-2673a-9105$ |
| 27.2-b4 |
27.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{11} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.913279723$ |
$3.284934771$ |
1.609423347 |
\( \frac{33550495500962}{81} a + \frac{114243625737751}{81} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -2698 a - 9184\) , \( -155890 a - 530828\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2698a-9184\right){x}-155890a-530828$ |
| 27.2-c1 |
27.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{10} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.109516830$ |
$17.20222275$ |
1.929703145 |
\( \frac{18631}{3} a - \frac{234592}{9} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 126 a - 562\) , \( -1479 a + 6510\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(126a-562\right){x}-1479a+6510$ |
| 27.2-d1 |
27.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{18} \) |
$1.59091$ |
$(a+3), (a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Ns |
$1$ |
\( 2^{2} \) |
$0.479224232$ |
$3.391723688$ |
1.664885244 |
\( -\frac{620650477}{729} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -568 a - 1932\) , \( 13533 a + 46080\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-568a-1932\right){x}+13533a+46080$ |
| 27.3-a1 |
27.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{3} \) |
$1.59091$ |
$(a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$28.65663988$ |
3.669106760 |
\( 218 a + 687 \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( a + 4\) , \( -a - 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+4\right){x}-a-1$ |
| 27.3-b1 |
27.3-b |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.3 |
\( 3^{3} \) |
\( - 3^{9} \) |
$1.59091$ |
$(a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$17.84899656$ |
2.285329830 |
\( 647332 a - 2738595 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 45 a - 200\) , \( -277 a + 1220\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(45a-200\right){x}-277a+1220$ |
| 27.3-c1 |
27.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.3 |
\( 3^{3} \) |
\( 3^{9} \) |
$1.59091$ |
$(a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$8.449141846$ |
1.081801760 |
\( 218 a + 687 \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 3 a + 6\) , \( 4 a + 12\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(3a+6\right){x}+4a+12$ |
| 27.3-d1 |
27.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.3 |
\( 3^{3} \) |
\( - 3^{3} \) |
$1.59091$ |
$(a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$6.301155290$ |
0.806780263 |
\( 647332 a - 2738595 \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( 283 a - 1230\) , \( 5006 a - 22030\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(283a-1230\right){x}+5006a-22030$ |
| 27.4-a1 |
27.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{3} \) |
$1.59091$ |
$(a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$28.65663988$ |
3.669106760 |
\( -218 a + 905 \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( a + 5\) , \( a + 3\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}+a+3$ |
| 27.4-b1 |
27.4-b |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.4 |
\( 3^{3} \) |
\( - 3^{9} \) |
$1.59091$ |
$(a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$17.84899656$ |
2.285329830 |
\( -647332 a - 2091263 \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -46 a - 154\) , \( 277 a + 943\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-46a-154\right){x}+277a+943$ |
| 27.4-c1 |
27.4-c |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.4 |
\( 3^{3} \) |
\( 3^{9} \) |
$1.59091$ |
$(a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$8.449141846$ |
1.081801760 |
\( -218 a + 905 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 5 a + 17\) , \( 9 a + 28\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5a+17\right){x}+9a+28$ |
| 27.4-d1 |
27.4-d |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
27.4 |
\( 3^{3} \) |
\( - 3^{3} \) |
$1.59091$ |
$(a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$6.301155290$ |
0.806780263 |
\( -647332 a - 2091263 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -275 a - 939\) , \( -6229 a - 21212\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-275a-939\right){x}-6229a-21212$ |
| 36.1-a1 |
36.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{12} \) |
$1.70954$ |
$(a+3), (a-4), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \cdot 5 \) |
$0.042952574$ |
$5.156018578$ |
2.268447579 |
\( -\frac{485109467}{944784} a - \frac{180850357}{59049} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -581 a - 1980\) , \( -19802 a - 67430\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-581a-1980\right){x}-19802a-67430$ |
| 36.1-b1 |
36.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{8} \) |
$1.70954$ |
$(a+3), (a-4), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$6.920108282$ |
2.658087219 |
\( -\frac{1409093}{17496} a + \frac{8240357}{17496} \) |
\( \bigl[a\) , \( a - 1\) , \( 1\) , \( 26 a + 91\) , \( -394 a - 1342\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(26a+91\right){x}-394a-1342$ |
| 36.1-c1 |
36.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{8} \) |
$1.70954$ |
$(a+3), (a-4), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$6.920108282$ |
2.658087219 |
\( \frac{1409093}{17496} a + \frac{284636}{729} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( -18 a + 125\) , \( 485 a - 2077\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(-18a+125\right){x}+485a-2077$ |
| 36.1-d1 |
36.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{61}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{12} \) |
$1.70954$ |
$(a+3), (a-4), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \cdot 5 \) |
$0.042952574$ |
$5.156018578$ |
2.268447579 |
\( \frac{485109467}{944784} a - \frac{1126238393}{314928} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 587 a - 2575\) , \( 17233 a - 75900\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(587a-2575\right){x}+17233a-75900$ |
*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.
