| 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 |
| 5.1-a1 |
5.1-a |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5^{3} \) |
$12.06803$ |
$(-a-1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.402545430$ |
$140.9706751$ |
2.387030356 |
\( \frac{44359379}{125} a^{3} - \frac{135124863}{125} a^{2} + \frac{54579778}{125} a + \frac{21745499}{125} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 4 a\) , \( a^{3} - a^{2} - 4 a + 3\) , \( -2 a^{2} + 8\) , \( -8 a^{3} + 9 a^{2} + 38 a - 29\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(-2a^{2}+8\right){x}-8a^{3}+9a^{2}+38a-29$ |
| 5.1-a2 |
5.1-a |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
5.1 |
\( 5 \) |
\( 5 \) |
$12.06803$ |
$(-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$4.207636291$ |
$1.740378705$ |
2.387030356 |
\( \frac{121609876762836773639}{5} a^{3} - \frac{370320600144419554323}{5} a^{2} + \frac{149312656151029707998}{5} a + \frac{59462502964063509694}{5} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 3\) , \( -a^{3} + 2 a^{2} + 6 a - 6\) , \( a^{2} - 2\) , \( 3 a^{3} - 35 a^{2} + 11 a + 10\) , \( -35 a^{3} - 223 a^{2} + 147 a + 47\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+6a-6\right){x}^{2}+\left(3a^{3}-35a^{2}+11a+10\right){x}-35a^{3}-223a^{2}+147a+47$ |
| 5.2-a1 |
5.2-a |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( -5 \) |
$12.06803$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$2.040924689$ |
$1.059299321$ |
1.957578386 |
\( -\frac{406845435648033265632}{5} a^{3} - \frac{596909700029974687497}{5} a^{2} + \frac{561551674365284296926}{5} a + \frac{164904035758107456836}{5} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a^{2} + a + 2\) , \( a^{2} - 2\) , \( 12 a^{3} - 143 a^{2} - 387 a - 84\) , \( -451 a^{3} - 2141 a^{2} - 2780 a - 594\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(12a^{3}-143a^{2}-387a-84\right){x}-451a^{3}-2141a^{2}-2780a-594$ |
| 5.2-a2 |
5.2-a |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
5.2 |
\( 5 \) |
\( - 5^{5} \) |
$12.06803$ |
$(a-2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$0.408184937$ |
$662.0620757$ |
1.957578386 |
\( -\frac{1215127}{3125} a^{3} - \frac{11206222}{3125} a^{2} + \frac{12386646}{3125} a + \frac{3520346}{3125} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a^{2} + a + 2\) , \( a^{2} - 2\) , \( 2 a^{3} - 3 a^{2} - 12 a + 11\) , \( a^{3} - 2 a^{2} - 5 a + 4\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(2a^{3}-3a^{2}-12a+11\right){x}+a^{3}-2a^{2}-5a+4$ |
| 17.1-a1 |
17.1-a |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$14.06278$ |
$(a^2-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1460.214306$ |
1.469087210 |
\( -\frac{10260436}{17} a^{3} + \frac{17034370}{17} a^{2} + \frac{53436493}{17} a - \frac{49599329}{17} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - 2 a^{2} - 5 a + 6\) , \( a^{3} - a^{2} - 5 a + 3\) , \( 4 a^{3} - 7 a^{2} - 21 a + 20\) , \( a^{3} - 5 a^{2} - 9 a + 11\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+6\right){x}^{2}+\left(4a^{3}-7a^{2}-21a+20\right){x}+a^{3}-5a^{2}-9a+11$ |
| 17.1-a2 |
17.1-a |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$14.06278$ |
$(a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$18.02733712$ |
1.469087210 |
\( \frac{1600853332539300377562}{4913} a^{3} + \frac{2348717987948992423862}{4913} a^{2} - \frac{2209588901430580983668}{4913} a - \frac{648863176367762796677}{4913} \) |
\( \bigl[a\) , \( -a^{3} + 4 a + 2\) , \( a^{2} - 3\) , \( 20 a^{3} - 30 a^{2} - 101 a + 95\) , \( 1065 a^{3} - 1312 a^{2} - 5023 a + 4326\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+4a+2\right){x}^{2}+\left(20a^{3}-30a^{2}-101a+95\right){x}+1065a^{3}-1312a^{2}-5023a+4326$ |
| 17.1-b1 |
17.1-b |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$14.06278$ |
$(a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$21.94066485$ |
1.787992864 |
\( \frac{2333477937110405}{4913} a^{3} - \frac{2895442533352176}{4913} a^{2} - \frac{10974038832479449}{4913} a + \frac{9656262537790317}{4913} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( 0\) , \( a^{2} + a - 2\) , \( -4 a^{3} + 12 a^{2} + 28 a - 28\) , \( -43 a^{3} + 21 a^{2} + 162 a - 125\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-4a^{3}+12a^{2}+28a-28\right){x}-43a^{3}+21a^{2}+162a-125$ |
| 17.1-b2 |
17.1-b |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$14.06278$ |
$(a^2-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1777.193853$ |
1.787992864 |
\( \frac{44405}{17} a^{3} - \frac{53172}{17} a^{2} - \frac{222413}{17} a + \frac{192484}{17} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( 0\) , \( a^{2} + a - 2\) , \( a^{3} - 3 a^{2} - 7 a + 7\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}-3a^{2}-7a+7\right){x}$ |
| 17.1-c1 |
17.1-c |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$14.06278$ |
$(a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.020445258$ |
$3336.028928$ |
2.470335267 |
\( \frac{2812482}{17} a^{3} - \frac{436671}{17} a^{2} - \frac{14622282}{17} a - \frac{3472740}{17} \) |
\( \bigl[1\) , \( -1\) , \( a^{3} - a^{2} - 5 a + 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - a^{2} - 5 a + 3\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-a^{2}-5a+2\right){y}={x}^{3}-{x}^{2}+\left(-a^{3}+a^{2}+5a-3\right){x}+a^{3}-a^{2}-5a+3$ |
| 17.1-d1 |
17.1-d |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{7} \) |
$14.06278$ |
$(a^2-a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 7 \) |
$0.039481750$ |
$193.8554219$ |
1.940465255 |
\( \frac{14550329350791}{410338673} a^{3} - \frac{17460526913829}{410338673} a^{2} - \frac{70013278651562}{410338673} a + \frac{59890133123066}{410338673} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( a^{2} - 3\) , \( 1\) , \( 2 a^{3} - 2 a^{2} - 2 a + 2\) , \( 4 a^{2} - 4 a - 2\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(2a^{3}-2a^{2}-2a+2\right){x}+4a^{2}-4a-2$ |
| 17.1-e1 |
17.1-e |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$14.06278$ |
$(a^2-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$1219.299900$ |
1.226708902 |
\( -\frac{13138941}{4913} a^{3} + \frac{55688846}{4913} a^{2} - \frac{47108313}{4913} a - \frac{6349259}{4913} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + 2 a^{2} + 4 a - 5\) , \( a^{3} - a^{2} - 5 a + 3\) , \( 5 a^{3} - 4 a^{2} - 11 a + 8\) , \( -a^{3} + 8 a^{2} + a - 6\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-5\right){x}^{2}+\left(5a^{3}-4a^{2}-11a+8\right){x}-a^{3}+8a^{2}+a-6$ |
| 17.1-e2 |
17.1-e |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{9} \) |
$14.06278$ |
$(a^2-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$15.05308519$ |
1.226708902 |
\( \frac{171160447031060120009}{118587876497} a^{3} + \frac{251122320658839236482}{118587876497} a^{2} - \frac{236244910976763578219}{118587876497} a - \frac{69376559071832157288}{118587876497} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + 2 a^{2} + 4 a - 5\) , \( a^{3} - a^{2} - 5 a + 3\) , \( 10 a^{3} - 39 a^{2} + 9 a + 13\) , \( 54 a^{3} - 287 a^{2} + 141 a + 47\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-5\right){x}^{2}+\left(10a^{3}-39a^{2}+9a+13\right){x}+54a^{3}-287a^{2}+141a+47$ |
| 25.2-a1 |
25.2-a |
$8$ |
$12$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{15} \) |
$14.75732$ |
$(-a-1), (a-2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.960528994$ |
$139.3246856$ |
3.330914597 |
\( \frac{37017593998106964766046}{244140625} a^{3} - \frac{45967160556571165951987}{244140625} a^{2} - \frac{173974709623222543685428}{244140625} a + \frac{153113815038849071118051}{244140625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 4 a\) , \( -24 a^{3} + 28 a^{2} + 111 a - 102\) , \( 154 a^{3} - 224 a^{2} - 628 a + 580\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(-24a^{3}+28a^{2}+111a-102\right){x}+154a^{3}-224a^{2}-628a+580$ |
| 25.2-a2 |
25.2-a |
$8$ |
$12$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{12} \) |
$14.75732$ |
$(-a-1), (a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.980264497$ |
$557.2987425$ |
3.330914597 |
\( \frac{84953323317}{15625} a^{3} - \frac{20801485241}{3125} a^{2} - \frac{397829716471}{15625} a + \frac{69849611116}{3125} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 4 a\) , \( a^{3} - 7 a^{2} + 6 a - 2\) , \( 10 a^{3} - 25 a^{2} - 8 a + 16\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(a^{3}-7a^{2}+6a-2\right){x}+10a^{3}-25a^{2}-8a+16$ |
| 25.2-a3 |
25.2-a |
$8$ |
$12$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{37} \) |
$14.75732$ |
$(-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$11.88158698$ |
$1.720057847$ |
3.330914597 |
\( \frac{46244668008223588302134090380476}{14551915228366851806640625} a^{3} + \frac{12490750472524989348389186178178}{14551915228366851806640625} a^{2} - \frac{33036313881222266037144554485668}{14551915228366851806640625} a - \frac{6709649267336155763899100497119}{14551915228366851806640625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 4 a\) , \( 126 a^{3} - 577 a^{2} + 371 a - 2\) , \( 4355 a^{3} - 14846 a^{2} + 5563 a + 2988\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(126a^{3}-577a^{2}+371a-2\right){x}+4355a^{3}-14846a^{2}+5563a+2988$ |
| 25.2-a4 |
25.2-a |
$8$ |
$12$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{6} \) |
$14.75732$ |
$(-a-1), (a-2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$0.990132248$ |
$1114.597485$ |
3.330914597 |
\( -\frac{2001543}{125} a^{3} + \frac{507398}{125} a^{2} + \frac{10175804}{125} a + \frac{1828741}{125} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 4 a\) , \( a^{3} - 2 a^{2} - 4 a + 3\) , \( -a - 1\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(a^{3}-2a^{2}-4a+3\right){x}-a-1$ |
| 25.2-a5 |
25.2-a |
$8$ |
$12$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{10} \) |
$14.75732$ |
$(-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$2.970396745$ |
$13.76046277$ |
3.330914597 |
\( -\frac{161349646797745637448}{1953125} a^{3} + \frac{29083069872157898656}{1953125} a^{2} + \frac{830589103931901589864}{1953125} a + \frac{196827557215524202237}{1953125} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 4 a\) , \( 16 a^{3} - 37 a^{2} - 9 a + 8\) , \( 79 a^{3} - 226 a^{2} + 65 a + 24\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(16a^{3}-37a^{2}-9a+8\right){x}+79a^{3}-226a^{2}+65a+24$ |
| 25.2-a6 |
25.2-a |
$8$ |
$12$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{15} \) |
$14.75732$ |
$(-a-1), (a-2)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$3.960528994$ |
$69.66234281$ |
3.330914597 |
\( \frac{5337284624243008162}{244140625} a^{3} + \frac{8077228533662853107}{244140625} a^{2} - \frac{7509324513679440076}{244140625} a - \frac{2212186689078341651}{244140625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 4 a\) , \( 26 a^{3} - 122 a^{2} + 61 a + 18\) , \( 446 a^{3} - 1206 a^{2} + 444 a + 200\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(26a^{3}-122a^{2}+61a+18\right){x}+446a^{3}-1206a^{2}+444a+200$ |
| 25.2-a7 |
25.2-a |
$8$ |
$12$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{20} \) |
$14.75732$ |
$(-a-1), (a-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$5.940793491$ |
$6.880231389$ |
3.330914597 |
\( -\frac{61665868033812302606224116}{3814697265625} a^{3} + \frac{187781917642534932622802152}{3814697265625} a^{2} - \frac{75713279102409741182559812}{3814697265625} a - \frac{30152186143757605919636971}{3814697265625} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 4 a\) , \( 181 a^{3} - 557 a^{2} + 221 a + 78\) , \( 4620 a^{3} - 14088 a^{2} + 5648 a + 2261\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(181a^{3}-557a^{2}+221a+78\right){x}+4620a^{3}-14088a^{2}+5648a+2261$ |
| 25.2-a8 |
25.2-a |
$8$ |
$12$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{13} \) |
$14.75732$ |
$(-a-1), (a-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$11.88158698$ |
$0.860028923$ |
3.330914597 |
\( -\frac{11898661225528847908779507575075052}{1953125} a^{3} + \frac{36233236010477647621295824993795094}{1953125} a^{2} - \frac{14609181092171472295526264936452364}{1953125} a - \frac{5817982858178861218692070233127237}{1953125} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 6 a - 5\) , \( 1\) , \( 2622 a^{3} - 556 a^{2} - 13287 a - 3210\) , \( 141890 a^{3} - 26609 a^{2} - 728076 a - 172581\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+6a-5\right){x}^{2}+\left(2622a^{3}-556a^{2}-13287a-3210\right){x}+141890a^{3}-26609a^{2}-728076a-172581$ |
| 25.3-a1 |
25.3-a |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{3} \) |
$14.75732$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.2 |
$1$ |
\( 2 \) |
$6.938541506$ |
$7.302502422$ |
3.670315310 |
\( -919628841950843 a^{3} - 1349248403596685 a^{2} + 1269324079417612 a + 372747008933503 \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - 4 a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -a^{3} + 3 a^{2} - a - 5\) , \( -a^{3} - 4 a^{2} + 6 a - 4\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(-a^{3}+3a^{2}-a-5\right){x}-a^{3}-4a^{2}+6a-4$ |
| 25.3-a2 |
25.3-a |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{3} \) |
$14.75732$ |
$(a-2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.1 |
$1$ |
\( 2 \) |
$1.387708301$ |
$912.8128028$ |
3.670315310 |
\( 34675 a^{3} - 105849 a^{2} + 42722 a + 17010 \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( a\) , \( a^{2} + a - 2\) , \( 2 a^{3} - 3 a^{2} - 10 a + 10\) , \( 10 a^{3} - 15 a^{2} - 48 a + 50\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+a{x}^{2}+\left(2a^{3}-3a^{2}-10a+10\right){x}+10a^{3}-15a^{2}-48a+50$ |
| 25.3-b1 |
25.3-b |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{4} \) |
$14.75732$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$0.545922001$ |
$158.3762906$ |
3.131513180 |
\( 199756208296625 a^{3} - 248050233034002 a^{2} - 938811137978626 a + 826240100450250 \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + a^{2} + 6 a - 3\) , \( a^{2} - 2\) , \( -33 a^{3} + 51 a^{2} + 152 a - 184\) , \( 301 a^{3} - 386 a^{2} - 1414 a + 1296\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a-3\right){x}^{2}+\left(-33a^{3}+51a^{2}+152a-184\right){x}+301a^{3}-386a^{2}-1414a+1296$ |
| 25.3-b2 |
25.3-b |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{8} \) |
$14.75732$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$0.109184400$ |
$791.8814532$ |
3.131513180 |
\( -1136 a^{3} - 2249 a^{2} + 794 a + 2276 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 5 a\) , \( a\) , \( -6 a^{3} + 3 a^{2} + 29 a - 4\) , \( -a^{3} - 2 a^{2} + 6 a + 11\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-6a^{3}+3a^{2}+29a-4\right){x}-a^{3}-2a^{2}+6a+11$ |
| 25.3-c1 |
25.3-c |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{9} \) |
$14.75732$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.110897772$ |
$319.1818356$ |
2.564038049 |
\( -14588 a^{3} - 20368 a^{2} + 21901 a + 6177 \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( a^{3} - 2 a^{2} - 5 a + 4\) , \( a^{3} - 4 a - 1\) , \( 4 a^{3} - 7 a^{2} - 22 a + 20\) , \( 7 a^{3} - a^{2} - 24 a + 13\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+4\right){x}^{2}+\left(4a^{3}-7a^{2}-22a+20\right){x}+7a^{3}-a^{2}-24a+13$ |
| 25.3-d1 |
25.3-d |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{11} \) |
$14.75732$ |
$(a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$158.5211894$ |
2.870719809 |
\( -\frac{1215127}{3125} a^{3} - \frac{11206222}{3125} a^{2} + \frac{12386646}{3125} a + \frac{3520346}{3125} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + a + 3\) , \( a^{3} - 4 a\) , \( 2 a^{3} - 3 a^{2} - 14 a + 4\) , \( -a^{2} - 6 a - 9\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(2a^{3}-3a^{2}-14a+4\right){x}-a^{2}-6a-9$ |
| 25.3-d2 |
25.3-d |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{7} \) |
$14.75732$ |
$(a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$25$ |
\( 2 \) |
$1$ |
$6.340847579$ |
2.870719809 |
\( -\frac{406845435648033265632}{5} a^{3} - \frac{596909700029974687497}{5} a^{2} + \frac{561551674365284296926}{5} a + \frac{164904035758107456836}{5} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a^{2} - a + 4\) , \( a^{3} - 5 a - 1\) , \( -202 a^{3} + 660 a^{2} - 354 a - 127\) , \( 57984 a^{3} - 176701 a^{2} + 71476 a + 28438\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-202a^{3}+660a^{2}-354a-127\right){x}+57984a^{3}-176701a^{2}+71476a+28438$ |
| 25.3-e1 |
25.3-e |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{3} \) |
$14.75732$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.031627943$ |
$1675.781968$ |
3.839299043 |
\( -14588 a^{3} - 20368 a^{2} + 21901 a + 6177 \) |
\( \bigl[a^{2} + a - 3\) , \( a^{3} - 2 a^{2} - 5 a + 6\) , \( a^{3} - 5 a\) , \( 7 a^{3} - 8 a^{2} - 34 a + 25\) , \( 7 a^{3} - 8 a^{2} - 33 a + 25\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+6\right){x}^{2}+\left(7a^{3}-8a^{2}-34a+25\right){x}+7a^{3}-8a^{2}-33a+25$ |
| 25.3-f1 |
25.3-f |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{2} \) |
$14.75732$ |
$(a-2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.1 |
$4$ |
\( 1 \) |
$1$ |
$1281.771328$ |
1.856966305 |
\( -1136 a^{3} - 2249 a^{2} + 794 a + 2276 \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a + 1\) , \( a^{3} - a^{2} - 4 a + 3\) , \( -5 a^{3} + 7 a^{2} + 14 a + 2\) , \( -7 a^{3} + 13 a^{2} + 12 a - 2\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-5a^{3}+7a^{2}+14a+2\right){x}-7a^{3}+13a^{2}+12a-2$ |
| 25.3-f2 |
25.3-f |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{10} \) |
$14.75732$ |
$(a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.1.2 |
$100$ |
\( 1 \) |
$1$ |
$2.050834125$ |
1.856966305 |
\( 199756208296625 a^{3} - 248050233034002 a^{2} - 938811137978626 a + 826240100450250 \) |
\( \bigl[a^{3} - 4 a\) , \( a^{2} - 4\) , \( a\) , \( 184 a^{3} + 278 a^{2} - 235 a - 94\) , \( -25251 a^{3} - 37044 a^{2} + 34874 a + 10215\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(184a^{3}+278a^{2}-235a-94\right){x}-25251a^{3}-37044a^{2}+34874a+10215$ |
| 25.3-g1 |
25.3-g |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{9} \) |
$14.75732$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$0.273990253$ |
$150.3111247$ |
2.983250439 |
\( 34675 a^{3} - 105849 a^{2} + 42722 a + 17010 \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 2 a^{2} + 5 a - 6\) , \( a^{3} - 5 a - 1\) , \( 19 a^{3} - 50 a^{2} + 6 a + 13\) , \( -112 a^{3} + 344 a^{2} - 140 a - 60\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-6\right){x}^{2}+\left(19a^{3}-50a^{2}+6a+13\right){x}-112a^{3}+344a^{2}-140a-60$ |
| 25.3-g2 |
25.3-g |
$2$ |
$5$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{9} \) |
$14.75732$ |
$(a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$1.369951265$ |
$30.06222494$ |
2.983250439 |
\( -919628841950843 a^{3} - 1349248403596685 a^{2} + 1269324079417612 a + 372747008933503 \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 6 a + 1\) , \( a^{3} - a^{2} - 4 a + 3\) , \( -16 a^{3} + 13 a^{2} + 68 a - 51\) , \( 33 a^{3} - 70 a^{2} - 189 a + 187\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(-a^{3}+6a+1\right){x}^{2}+\left(-16a^{3}+13a^{2}+68a-51\right){x}+33a^{3}-70a^{2}-189a+187$ |
| 25.4-a1 |
25.4-a |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.4 |
\( 5^{2} \) |
\( 5^{7} \) |
$14.75732$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$9$ |
\( 2^{2} \) |
$1$ |
$6.554615523$ |
2.136599935 |
\( \frac{121609876762836773639}{5} a^{3} - \frac{370320600144419554323}{5} a^{2} + \frac{149312656151029707998}{5} a + \frac{59462502964063509694}{5} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 6 a + 2\) , \( a^{3} - 4 a\) , \( 250 a^{3} - 76 a^{2} - 1206 a - 284\) , \( -2636 a^{3} + 284 a^{2} + 14017 a + 3332\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+6a+2\right){x}^{2}+\left(250a^{3}-76a^{2}-1206a-284\right){x}-2636a^{3}+284a^{2}+14017a+3332$ |
| 25.4-a2 |
25.4-a |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
25.4 |
\( 5^{2} \) |
\( 5^{9} \) |
$14.75732$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$1$ |
$58.99153970$ |
2.136599935 |
\( \frac{44359379}{125} a^{3} - \frac{135124863}{125} a^{2} + \frac{54579778}{125} a + \frac{21745499}{125} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 6 a + 2\) , \( a^{3} - 4 a\) , \( -10 a^{3} + 4 a^{2} + 59 a + 16\) , \( -42 a^{3} + 13 a^{2} + 229 a + 54\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+6a+2\right){x}^{2}+\left(-10a^{3}+4a^{2}+59a+16\right){x}-42a^{3}+13a^{2}+229a+54$ |
| 37.1-a1 |
37.1-a |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$15.49852$ |
$(a^3-a^2-3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.066782126$ |
$1115.440444$ |
2.697988619 |
\( \frac{786862291}{37} a^{3} + \frac{1154463483}{37} a^{2} - \frac{1086041587}{37} a - \frac{318918004}{37} \) |
\( \bigl[a\) , \( -a^{3} + a^{2} + 6 a - 3\) , \( a\) , \( -2 a^{2} + 7\) , \( -2 a^{3} + a^{2} + 10 a - 2\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a^{3}+a^{2}+6a-3\right){x}^{2}+\left(-2a^{2}+7\right){x}-2a^{3}+a^{2}+10a-2$ |
| 37.1-b1 |
37.1-b |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$15.49852$ |
$(a^3-a^2-3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.296531769$ |
$418.7900179$ |
4.497809218 |
\( \frac{582072}{37} a^{3} + \frac{74577}{37} a^{2} - \frac{3409385}{37} a - \frac{741191}{37} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{2} - 3\) , \( a\) , \( 3 a^{3} + 4 a^{2} - 5 a - 1\) , \( 5 a^{3} + 7 a^{2} - 7 a - 2\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(3a^{3}+4a^{2}-5a-1\right){x}+5a^{3}+7a^{2}-7a-2$ |
| 37.1-c1 |
37.1-c |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( 37 \) |
$15.49852$ |
$(a^3-a^2-3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$2.884214457$ |
$41.00427629$ |
4.283415265 |
\( \frac{911719667}{37} a^{3} - \frac{5270784926}{37} a^{2} - \frac{9633998057}{37} a + \frac{10915255385}{37} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} - 2 a^{2} - 6 a + 5\) , \( 1\) , \( 6 a^{3} - 11 a^{2} - 16 a + 8\) , \( 12 a^{3} - 25 a^{2} - 15 a + 4\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+{y}={x}^{3}+\left(a^{3}-2a^{2}-6a+5\right){x}^{2}+\left(6a^{3}-11a^{2}-16a+8\right){x}+12a^{3}-25a^{2}-15a+4$ |
| 37.1-d1 |
37.1-d |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
37.1 |
\( 37 \) |
\( -37 \) |
$15.49852$ |
$(a^3-a^2-3a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$220.8642298$ |
1.999856681 |
\( \frac{58867147802}{37} a^{3} - \frac{10610346225}{37} a^{2} - \frac{303033963992}{37} a - \frac{71812707135}{37} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} - a^{2} - 5 a + 3\) , \( -2 a^{3} + 3 a^{2} + 9 a - 10\) , \( 6 a^{3} - 11 a^{2} - 33 a + 30\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(-2a^{3}+3a^{2}+9a-10\right){x}+6a^{3}-11a^{2}-33a+30$ |
| 37.2-a1 |
37.2-a |
$3$ |
$9$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
37.2 |
\( 37 \) |
\( - 37^{3} \) |
$15.49852$ |
$(a^3-4a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$81$ |
\( 3 \) |
$1.868714284$ |
$0.146243614$ |
2.405246386 |
\( -\frac{2661644356519596892007238}{50653} a^{3} + \frac{482699095303740366589127}{50653} a^{2} + \frac{13700262782894735564670592}{50653} a + \frac{3232052330066604775157553}{50653} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + 6 a\) , \( a^{3} - a^{2} - 5 a + 3\) , \( 51 a^{3} + 180 a^{2} - 356 a - 1163\) , \( 1345 a^{3} + 2407 a^{2} - 8251 a - 16063\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}+\left(-a^{3}+6a\right){x}^{2}+\left(51a^{3}+180a^{2}-356a-1163\right){x}+1345a^{3}+2407a^{2}-8251a-16063$ |
| 37.2-a2 |
37.2-a |
$3$ |
$9$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
37.2 |
\( 37 \) |
\( - 37^{9} \) |
$15.49852$ |
$(a^3-4a-2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$9$ |
\( 3^{2} \) |
$0.622904761$ |
$11.84573277$ |
2.405246386 |
\( -\frac{182667213631522428976}{129961739795077} a^{3} + \frac{50401070034046346530}{129961739795077} a^{2} + \frac{932928939055723203988}{129961739795077} a + \frac{134727608330513111571}{129961739795077} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + 6 a\) , \( a^{3} - a^{2} - 5 a + 3\) , \( a^{3} + 5 a^{2} - 6 a - 8\) , \( 5 a^{3} + 6 a^{2} - 25 a - 34\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}+\left(-a^{3}+6a\right){x}^{2}+\left(a^{3}+5a^{2}-6a-8\right){x}+5a^{3}+6a^{2}-25a-34$ |
| 37.2-a3 |
37.2-a |
$3$ |
$9$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
37.2 |
\( 37 \) |
\( - 37^{3} \) |
$15.49852$ |
$(a^3-4a-2)$ |
$1$ |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1.868714284$ |
$959.5043551$ |
2.405246386 |
\( \frac{204498960}{50653} a^{3} - \frac{700450852}{50653} a^{2} + \frac{417659607}{50653} a + \frac{155795785}{50653} \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} + 6 a\) , \( a^{3} - a^{2} - 5 a + 3\) , \( a^{3} - a + 7\) , \( -a^{3} + 3 a^{2} + 6 a - 3\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-a^{2}-5a+3\right){y}={x}^{3}+\left(-a^{3}+6a\right){x}^{2}+\left(a^{3}-a+7\right){x}-a^{3}+3a^{2}+6a-3$ |
| 41.1-a1 |
41.1-a |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41 \) |
$15.69868$ |
$(a^3-a^2-5a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$371.1581134$ |
3.360720898 |
\( \frac{48418968}{41} a^{3} - \frac{8711458}{41} a^{2} - 6079306 a - \frac{59137239}{41} \) |
\( \bigl[a^{3} - 5 a\) , \( a^{3} - a^{2} - 5 a + 3\) , \( a^{3} - a^{2} - 4 a + 3\) , \( -3 a^{3} + 4 a^{2} + 14 a - 12\) , \( -15 a^{3} + 18 a^{2} + 70 a - 61\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+3\right){x}^{2}+\left(-3a^{3}+4a^{2}+14a-12\right){x}-15a^{3}+18a^{2}+70a-61$ |
| 41.1-b1 |
41.1-b |
$2$ |
$2$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( 41 \) |
$15.69868$ |
$(a^3-a^2-5a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$725.7869111$ |
1.642943500 |
\( \frac{15678507664}{41} a^{3} - \frac{19528962612}{41} a^{2} - 1798976020 a + \frac{64949959549}{41} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( a^{3} - 2 a^{2} - 5 a + 4\) , \( a^{3} - 4 a\) , \( -13 a^{3} - 23 a^{2} + 14 a + 11\) , \( 118 a^{3} + 175 a^{2} - 163 a - 48\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-5a+4\right){x}^{2}+\left(-13a^{3}-23a^{2}+14a+11\right){x}+118a^{3}+175a^{2}-163a-48$ |
| 41.1-b2 |
41.1-b |
$2$ |
$2$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
41.1 |
\( 41 \) |
\( - 41^{2} \) |
$15.69868$ |
$(a^3-a^2-5a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$362.8934555$ |
1.642943500 |
\( \frac{9775763060507010}{1681} a^{3} + \frac{14344201751225366}{1681} a^{2} - \frac{329120734294446}{41} a - \frac{3962648359812209}{1681} \) |
\( \bigl[a^{3} - 5 a\) , \( a^{3} - 6 a - 2\) , \( a^{3} - a^{2} - 5 a + 2\) , \( -123 a^{3} + 136 a^{2} + 558 a - 482\) , \( -1092 a^{3} + 1430 a^{2} + 5222 a - 4642\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{3}-a^{2}-5a+2\right){y}={x}^{3}+\left(a^{3}-6a-2\right){x}^{2}+\left(-123a^{3}+136a^{2}+558a-482\right){x}-1092a^{3}+1430a^{2}+5222a-4642$ |
| 55.1-a1 |
55.1-a |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11^{2} \) |
$16.28585$ |
$(a-2), (-a^3+a^2+4a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$122.9570791$ |
2.226675966 |
\( -\frac{5701048718}{605} a^{3} - \frac{8272058303}{605} a^{2} + \frac{8014199619}{605} a + \frac{2220711199}{605} \) |
\( \bigl[a^{3} - a^{2} - 5 a + 3\) , \( -a^{3} + 4 a + 1\) , \( a^{2} + a - 3\) , \( 4 a^{3} - 4 a^{2} - 16 a + 9\) , \( -a^{3} + a^{2}\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-5a+3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(4a^{3}-4a^{2}-16a+9\right){x}-a^{3}+a^{2}$ |
| 55.1-b1 |
55.1-b |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5 \cdot 11^{18} \) |
$16.28585$ |
$(a-2), (-a^3+a^2+4a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \) |
$1.322324109$ |
$4.918316152$ |
4.239946169 |
\( -\frac{441188499593029758855806480218}{27799586567461157405} a^{3} + \frac{79523587355841257141300433142}{27799586567461157405} a^{2} + \frac{2271131974155936637959764742974}{27799586567461157405} a + \frac{538197967364568859678163986909}{27799586567461157405} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 6 a + 2\) , \( a^{2} + a - 3\) , \( 168 a^{3} - 528 a^{2} + 235 a + 96\) , \( -7993 a^{3} + 24348 a^{2} - 9849 a - 3906\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+6a+2\right){x}^{2}+\left(168a^{3}-528a^{2}+235a+96\right){x}-7993a^{3}+24348a^{2}-9849a-3906$ |
| 55.1-b2 |
55.1-b |
$2$ |
$3$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( - 5^{3} \cdot 11^{6} \) |
$16.28585$ |
$(a-2), (-a^3+a^2+4a-2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.440774703$ |
$398.3836083$ |
4.239946169 |
\( \frac{20642606956146187}{221445125} a^{3} - \frac{62907934461514993}{221445125} a^{2} + \frac{25502466358337349}{221445125} a + \frac{9997767950770274}{221445125} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - 2 a^{2} - 6 a + 4\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 42 a^{3} - 16 a^{2} - 195 a - 33\) , \( 131 a^{3} - 9 a^{2} - 717 a - 161\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-2a^{2}-6a+4\right){x}^{2}+\left(42a^{3}-16a^{2}-195a-33\right){x}+131a^{3}-9a^{2}-717a-161$ |
| 55.1-c1 |
55.1-c |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
55.1 |
\( 5 \cdot 11 \) |
\( 5 \cdot 11^{4} \) |
$16.28585$ |
$(a-2), (-a^3+a^2+4a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.072124268$ |
$396.9723880$ |
4.147966012 |
\( -\frac{4600529}{73205} a^{3} + \frac{17232266}{73205} a^{2} + \frac{2753472}{73205} a - \frac{396018}{73205} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - 5 a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -2 a^{3} + a^{2} + 10 a\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(-2a^{3}+a^{2}+10a\right){x}$ |
| 55.2-a1 |
55.2-a |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
55.2 |
\( 5 \cdot 11 \) |
\( 5 \cdot 11^{2} \) |
$16.28585$ |
$(-a-1), (-a^3+a^2+4a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.073583425$ |
$509.8618549$ |
2.717665475 |
\( -\frac{615811}{605} a^{3} - \frac{3814643}{605} a^{2} + \frac{1021133}{605} a + \frac{1151444}{605} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{2} + 2\) , \( a^{3} - a^{2} - 4 a + 3\) , \( -3 a^{3} - a^{2} + 12 a + 5\) , \( -3 a^{3} + 13 a\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-3a^{3}-a^{2}+12a+5\right){x}-3a^{3}+13a$ |
| 65.1-a1 |
65.1-a |
$1$ |
$1$ |
4.4.12197.1 |
$4$ |
$[4, 0]$ |
65.1 |
\( 5 \cdot 13 \) |
\( 5^{6} \cdot 13^{7} \) |
$16.62950$ |
$(a-2), (-a^2+a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$55.96311236$ |
1.013457038 |
\( \frac{124626896658386003}{980445578125} a^{3} - \frac{37489721953370792}{980445578125} a^{2} - \frac{675916132450731469}{980445578125} a - \frac{159320769484119519}{980445578125} \) |
\( \bigl[a^{2} - 2\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{2} - 3\) , \( 31 a^{3} - 6 a^{2} - 160 a - 35\) , \( -166 a^{3} + 31 a^{2} + 854 a + 197\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(31a^{3}-6a^{2}-160a-35\right){x}-166a^{3}+31a^{2}+854a+197$ |
*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.
