| 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 |
| 9.2-a1 |
9.2-a |
$1$ |
$1$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{10} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$55.16484240$ |
0.496256100 |
\( \frac{110806447}{243} a^{3} - \frac{681926651}{243} a^{2} + \frac{33050716}{27} a + \frac{134090341}{81} \) |
\( \bigl[a + 1\) , \( a^{3} - 2 a^{2} - 4 a + 4\) , \( a^{3} - 3 a - 1\) , \( 21 a^{3} - 60 a^{2} + 6 a + 36\) , \( 125 a^{3} - 365 a^{2} + 83 a + 191\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+4\right){x}^{2}+\left(21a^{3}-60a^{2}+6a+36\right){x}+125a^{3}-365a^{2}+83a+191$ |
| 9.2-b1 |
9.2-b |
$1$ |
$1$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{10} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3^{2} \) |
$0.010391795$ |
$716.2286652$ |
2.410395343 |
\( \frac{110806447}{243} a^{3} - \frac{681926651}{243} a^{2} + \frac{33050716}{27} a + \frac{134090341}{81} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - 4 a\) , \( 15 a^{3} - 2 a^{2} - 71 a - 20\) , \( -50 a^{3} - 10 a^{2} + 241 a + 138\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(15a^{3}-2a^{2}-71a-20\right){x}-50a^{3}-10a^{2}+241a+138$ |
| 9.2-c1 |
9.2-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{12} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$739.8485530$ |
2.218528696 |
\( -\frac{2960660}{243} a^{3} - \frac{414782}{243} a^{2} + \frac{1579541}{27} a + \frac{2534209}{81} \) |
\( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} - a^{2} - 4 a + 3\) , \( 2 a^{2} - a\) , \( 4 a^{2} - 6\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(2a^{2}-a\right){x}+4a^{2}-6$ |
| 9.2-c2 |
9.2-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{12} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{3} \) |
$1$ |
$82.20539478$ |
2.218528696 |
\( -\frac{1437830}{243} a^{3} - \frac{328570}{243} a^{2} + \frac{6995050}{243} a + \frac{422507}{27} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - a^{2} - 5 a + 3\) , \( a^{3} - 4 a\) , \( 7 a^{3} - 15 a^{2} - 13 a + 21\) , \( 21 a^{3} - 57 a^{2} - 3 a + 45\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+3\right){x}^{2}+\left(7a^{3}-15a^{2}-13a+21\right){x}+21a^{3}-57a^{2}-3a+45$ |
| 9.2-c3 |
9.2-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{4} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$81$ |
\( 3 \) |
$1$ |
$1.014881417$ |
2.218528696 |
\( -\frac{72675578002462}{9} a^{3} - \frac{11674145137967}{9} a^{2} + \frac{349891522738418}{9} a + 20881581070793 \) |
\( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - a^{2} - 5 a + 3\) , \( a^{3} - 4 a\) , \( 302 a^{3} - 935 a^{2} + 242 a + 501\) , \( 9239 a^{3} - 27481 a^{2} + 7068 a + 14361\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+3\right){x}^{2}+\left(302a^{3}-935a^{2}+242a+501\right){x}+9239a^{3}-27481a^{2}+7068a+14361$ |
| 9.2-d1 |
9.2-d |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{16} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$586.2798230$ |
1.318525196 |
\( \frac{16187635}{81} a^{3} - \frac{128913686}{729} a^{2} - \frac{162464473}{243} a + \frac{207672871}{243} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 5\) , \( a^{3} - 3 a\) , \( 94 a^{3} - 278 a^{2} + 67 a + 150\) , \( -1115 a^{3} + 3285 a^{2} - 831 a - 1716\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-5\right){x}^{2}+\left(94a^{3}-278a^{2}+67a+150\right){x}-1115a^{3}+3285a^{2}-831a-1716$ |
| 9.2-d2 |
9.2-d |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{19} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$73.28497788$ |
1.318525196 |
\( \frac{293973114714786138876468353}{6561} a^{3} + \frac{397699404466429465133072960}{6561} a^{2} - \frac{534141402909392338155196883}{6561} a - \frac{374831396846942149724516810}{6561} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 5\) , \( a^{3} - 3 a\) , \( 1354 a^{3} - 4008 a^{2} + 1007 a + 2105\) , \( -96998 a^{3} + 286129 a^{2} - 72706 a - 149327\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-5\right){x}^{2}+\left(1354a^{3}-4008a^{2}+1007a+2105\right){x}-96998a^{3}+286129a^{2}-72706a-149327$ |
| 9.2-d3 |
9.2-d |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{14} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$586.2798230$ |
1.318525196 |
\( \frac{153280024570}{3} a^{3} + \frac{5949423007219}{81} a^{2} - \frac{6639850799911}{81} a - \frac{4892331544796}{81} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 5\) , \( a^{3} - 3 a\) , \( 1429 a^{3} - 4213 a^{2} + 1062 a + 2200\) , \( -87397 a^{3} + 257765 a^{2} - 65501 a - 134499\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-5\right){x}^{2}+\left(1429a^{3}-4213a^{2}+1062a+2200\right){x}-87397a^{3}+257765a^{2}-65501a-134499$ |
| 9.2-d4 |
9.2-d |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{7} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$293.1399115$ |
1.318525196 |
\( -\frac{252064245288220542115}{3} a^{3} - \frac{40398765753634897600}{3} a^{2} + \frac{3640340736570996037259}{9} a + \frac{1955210267140174098986}{9} \) |
\( \bigl[a^{3} - 4 a\) , \( -1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -442 a^{3} - 360 a^{2} + 1055 a - 59\) , \( 10644 a^{3} + 11834 a^{2} - 21761 a - 7303\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}-{x}^{2}+\left(-442a^{3}-360a^{2}+1055a-59\right){x}+10644a^{3}+11834a^{2}-21761a-7303$ |
| 9.2-d5 |
9.2-d |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{26} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$36.64248894$ |
1.318525196 |
\( \frac{98915472999725026}{531441} a^{3} - \frac{154676985561569491}{531441} a^{2} - \frac{135793825274986963}{177147} a + \frac{175466380177934948}{177147} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( a - 1\) , \( a^{3} - 4 a\) , \( -20 a^{3} + 30 a^{2} + 53 a - 156\) , \( -141 a^{3} + 261 a^{2} + 588 a - 891\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-20a^{3}+30a^{2}+53a-156\right){x}-141a^{3}+261a^{2}+588a-891$ |
| 9.2-d6 |
9.2-d |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{8} \) |
$13.07300$ |
$(a), (a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$586.2798230$ |
1.318525196 |
\( -\frac{3124}{27} a^{3} + \frac{3853}{27} a^{2} + \frac{2719}{9} a - \frac{3467}{9} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( a - 1\) , \( a^{3} - 4 a\) , \( -2 a - 1\) , \( a + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a-1\right){x}+a+1$ |
| 9.2-e1 |
9.2-e |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{12} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$0.057009731$ |
$1254.278923$ |
2.573040308 |
\( -\frac{1437830}{243} a^{3} - \frac{328570}{243} a^{2} + \frac{6995050}{243} a + \frac{422507}{27} \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( 3 a^{3} + 5 a^{2} - 5 a - 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(-a^{3}-a^{2}+3a+2\right){x}+3a^{3}+5a^{2}-5a-7$ |
| 9.2-e2 |
9.2-e |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{4} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$0.171029195$ |
$1254.278923$ |
2.573040308 |
\( -\frac{72675578002462}{9} a^{3} - \frac{11674145137967}{9} a^{2} + \frac{349891522738418}{9} a + 20881581070793 \) |
\( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -216 a^{3} - 291 a^{2} + 393 a + 257\) , \( 4164 a^{3} + 5639 a^{2} - 7559 a - 5290\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(-216a^{3}-291a^{2}+393a+257\right){x}+4164a^{3}+5639a^{2}-7559a-5290$ |
| 9.2-e3 |
9.2-e |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{12} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$0.171029195$ |
$139.3643247$ |
2.573040308 |
\( -\frac{2960660}{243} a^{3} - \frac{414782}{243} a^{2} + \frac{1579541}{27} a + \frac{2534209}{81} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a^{2} - a + 4\) , \( 1\) , \( 4 a^{3} - 4 a^{2} - 20 a + 21\) , \( 6 a^{3} - 9 a^{2} - 24 a + 26\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(4a^{3}-4a^{2}-20a+21\right){x}+6a^{3}-9a^{2}-24a+26$ |
| 9.2-f1 |
9.2-f |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{7} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \cdot 3 \) |
$3.396211229$ |
$0.910871761$ |
2.671570516 |
\( -\frac{252064245288220542115}{3} a^{3} - \frac{40398765753634897600}{3} a^{2} + \frac{3640340736570996037259}{9} a + \frac{1955210267140174098986}{9} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a + 2\) , \( a^{3} - 3 a\) , \( 248 a^{3} - 135 a^{2} - 894 a - 473\) , \( 3955 a^{3} - 1917 a^{2} - 14565 a - 7071\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(248a^{3}-135a^{2}-894a-473\right){x}+3955a^{3}-1917a^{2}-14565a-7071$ |
| 9.2-f2 |
9.2-f |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{26} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$0.424526403$ |
$233.1831710$ |
2.671570516 |
\( \frac{98915472999725026}{531441} a^{3} - \frac{154676985561569491}{531441} a^{2} - \frac{135793825274986963}{177147} a + \frac{175466380177934948}{177147} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a + 2\) , \( a^{3} - 3 a\) , \( -12 a^{3} + 15 a^{2} + 51 a - 63\) , \( 38 a^{3} - 60 a^{2} - 162 a + 201\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(-12a^{3}+15a^{2}+51a-63\right){x}+38a^{3}-60a^{2}-162a+201$ |
| 9.2-f3 |
9.2-f |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{14} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \cdot 3 \) |
$1.698105614$ |
$14.57394818$ |
2.671570516 |
\( \frac{153280024570}{3} a^{3} + \frac{5949423007219}{81} a^{2} - \frac{6639850799911}{81} a - \frac{4892331544796}{81} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a + 2\) , \( a^{3} - 3 a\) , \( 8 a^{3} - 15 a^{2} - 39 a - 23\) , \( 52 a^{3} - 78 a^{2} - 246 a - 105\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(8a^{3}-15a^{2}-39a-23\right){x}+52a^{3}-78a^{2}-246a-105$ |
| 9.2-f4 |
9.2-f |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{16} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.849052807$ |
$233.1831710$ |
2.671570516 |
\( \frac{16187635}{81} a^{3} - \frac{128913686}{729} a^{2} - \frac{162464473}{243} a + \frac{207672871}{243} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a + 2\) , \( a^{3} - 3 a\) , \( -2 a^{3} + 6 a - 3\) , \( a^{3} - 3 a^{2} - 6 a\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(-2a^{3}+6a-3\right){x}+a^{3}-3a^{2}-6a$ |
| 9.2-f5 |
9.2-f |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{8} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.424526403$ |
$233.1831710$ |
2.671570516 |
\( -\frac{3124}{27} a^{3} + \frac{3853}{27} a^{2} + \frac{2719}{9} a - \frac{3467}{9} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a + 2\) , \( a^{3} - 3 a\) , \( -2 a^{3} + 6 a + 2\) , \( -a^{3} + 3 a + 1\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(-2a^{3}+6a+2\right){x}-a^{3}+3a+1$ |
| 9.2-f6 |
9.2-f |
$6$ |
$8$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{19} \) |
$13.07300$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \cdot 3 \) |
$3.396211229$ |
$0.910871761$ |
2.671570516 |
\( \frac{293973114714786138876468353}{6561} a^{3} + \frac{397699404466429465133072960}{6561} a^{2} - \frac{534141402909392338155196883}{6561} a - \frac{374831396846942149724516810}{6561} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{3} + 3 a + 2\) , \( a^{3} - 3 a\) , \( -72 a^{3} - 135 a^{2} + 96 a + 107\) , \( -1327 a^{3} - 1959 a^{2} + 2253 a + 1681\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(-72a^{3}-135a^{2}+96a+107\right){x}-1327a^{3}-1959a^{2}+2253a+1681$ |
| 16.1-a1 |
16.1-a |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$3.848457779$ |
$1.923527708$ |
2.397348384 |
\( -\frac{573746144654542909099}{2} a^{3} + 448591720203712073410 a^{2} + 1181481234689977876243 a - \frac{3053308720069032929009}{2} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 6\) , \( a^{3} - 4 a - 1\) , \( -27 a^{3} + 38 a^{2} + 129 a - 152\) , \( -225 a^{3} + 384 a^{2} + 950 a - 1419\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+6\right){x}^{2}+\left(-27a^{3}+38a^{2}+129a-152\right){x}-225a^{3}+384a^{2}+950a-1419$ |
| 16.1-a2 |
16.1-a |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$1.282819259$ |
$155.8057444$ |
2.397348384 |
\( -\frac{3615673}{2} a^{3} + \frac{22615701}{8} a^{2} + \frac{14891109}{2} a - 9620794 \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 6\) , \( a^{3} - 4 a - 1\) , \( 3 a^{3} - 2 a^{2} - 11 a + 8\) , \( a^{3} - 2 a + 3\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+6\right){x}^{2}+\left(3a^{3}-2a^{2}-11a+8\right){x}+a^{3}-2a+3$ |
| 16.1-a3 |
16.1-a |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.427606419$ |
$1402.251699$ |
2.397348384 |
\( \frac{6596973}{2} a^{3} - \frac{3508603}{2} a^{2} - \frac{43232855}{2} a - \frac{22074779}{2} \) |
\( \bigl[a^{3} - 3 a\) , \( -a^{2} + 4\) , \( 0\) , \( a^{3} - 2 a + 5\) , \( -5 a^{3} + 9 a^{2} + 21 a - 27\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(a^{3}-2a+5\right){x}-5a^{3}+9a^{2}+21a-27$ |
| 16.1-b1 |
16.1-b |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$0 \le r \le 1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
|
\( 1 \) |
$1$ |
$475.9227258$ |
1.518617981 |
\( -\frac{573746144654542909099}{2} a^{3} + 448591720203712073410 a^{2} + 1181481234689977876243 a - \frac{3053308720069032929009}{2} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 4 a - 2\) , \( a^{2} + a - 3\) , \( -81 a^{3} + 102 a^{2} + 343 a - 318\) , \( -257 a^{3} - 912 a^{2} + 1591 a + 4663\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(-81a^{3}+102a^{2}+343a-318\right){x}-257a^{3}-912a^{2}+1591a+4663$ |
| 16.1-b2 |
16.1-b |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
|
\( 1 \) |
$1$ |
$52.88030287$ |
1.518617981 |
\( \frac{6596973}{2} a^{3} - \frac{3508603}{2} a^{2} - \frac{43232855}{2} a - \frac{22074779}{2} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{2} - 2\) , \( 0\) , \( 2 a^{3} + 4 a^{2} - 6 a - 4\) , \( 4 a^{3} - 2 a^{2} - 4 a\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(2a^{3}+4a^{2}-6a-4\right){x}+4a^{3}-2a^{2}-4a$ |
| 16.1-b3 |
16.1-b |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$14.04786$ |
$(2)$ |
$0 \le r \le 1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
|
\( 3 \) |
$1$ |
$475.9227258$ |
1.518617981 |
\( -\frac{3615673}{2} a^{3} + \frac{22615701}{8} a^{2} + \frac{14891109}{2} a - 9620794 \) |
\( \bigl[a^{3} - a^{2} - 3 a + 3\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( -a^{2} + 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(-a^{3}+a^{2}+3a-3\right){x}-a^{2}+1$ |
| 16.1-c1 |
16.1-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1.628139834$ |
$729.6575744$ |
4.749760984 |
\( \frac{54906875467771}{2} a^{3} + 77871175883534 a^{2} + 52479510197237 a + \frac{19511461081835}{2} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 2 a^{2} + 5 a - 6\) , \( a^{3} - 3 a - 1\) , \( -5 a^{3} - a^{2} + 15 a - 8\) , \( 2380 a^{3} - 3719 a^{2} - 9802 a + 12660\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+5a-6\right){x}^{2}+\left(-5a^{3}-a^{2}+15a-8\right){x}+2380a^{3}-3719a^{2}-9802a+12660$ |
| 16.1-c2 |
16.1-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1.628139834$ |
$81.07306383$ |
4.749760984 |
\( -1495210 a^{3} + \frac{4635579}{2} a^{2} + \frac{12430071}{2} a - 7948001 \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - 3 a\) , \( -2 a^{3} - 3 a^{2} + 3 a + 3\) , \( -4 a^{3} - 7 a^{2} + 3 a + 2\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-2a^{3}-3a^{2}+3a+3\right){x}-4a^{3}-7a^{2}+3a+2$ |
| 16.1-c3 |
16.1-c |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.542713278$ |
$729.6575744$ |
4.749760984 |
\( -\frac{6035}{2} a^{3} + \frac{59547}{8} a^{2} + \frac{135879}{4} a + \frac{64975}{4} \) |
\( \bigl[a\) , \( 1\) , \( a^{3} - 4 a - 1\) , \( 2 a^{3} - 10 a - 4\) , \( -8 a^{3} + 9 a^{2} + 34 a - 27\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+{x}^{2}+\left(2a^{3}-10a-4\right){x}-8a^{3}+9a^{2}+34a-27$ |
| 16.1-d1 |
16.1-d |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1.081497900$ |
$6.194235184$ |
2.169498472 |
\( \frac{54906875467771}{2} a^{3} + 77871175883534 a^{2} + 52479510197237 a + \frac{19511461081835}{2} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 4 a + 1\) , \( 0\) , \( 18 a^{3} - 81 a - 45\) , \( 73 a^{3} + 95 a^{2} - 370 a - 607\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(18a^{3}-81a-45\right){x}+73a^{3}+95a^{2}-370a-607$ |
| 16.1-d2 |
16.1-d |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{12} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3 \) |
$0.360499300$ |
$501.7330499$ |
2.169498472 |
\( -\frac{6035}{2} a^{3} + \frac{59547}{8} a^{2} + \frac{135879}{4} a + \frac{64975}{4} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 4 a + 1\) , \( 0\) , \( -2 a^{3} + 9 a + 5\) , \( a^{3} - 3 a^{2} - 4 a + 13\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(-2a^{3}+9a+5\right){x}+a^{3}-3a^{2}-4a+13$ |
| 16.1-d3 |
16.1-d |
$3$ |
$9$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
16.1 |
\( 2^{4} \) |
\( 2^{4} \) |
$14.04786$ |
$(2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$0.120166433$ |
$4515.597449$ |
2.169498472 |
\( -1495210 a^{3} + \frac{4635579}{2} a^{2} + \frac{12430071}{2} a - 7948001 \) |
\( \bigl[1\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a + 1\) , \( 10 a^{3} - 29 a^{2} + 3 a + 21\) , \( -51 a^{3} + 149 a^{2} - 35 a - 81\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(10a^{3}-29a^{2}+3a+21\right){x}-51a^{3}+149a^{2}-35a-81$ |
| 21.2-a1 |
21.2-a |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{12} \) |
$14.53358$ |
$(a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$134.5635571$ |
3.631551357 |
\( \frac{22865829504156704}{41523861603} a^{3} - \frac{34444842533329259}{41523861603} a^{2} - \frac{93278703923001637}{41523861603} a + \frac{39651957308694504}{13841287201} \) |
\( \bigl[a^{2} + a - 2\) , \( a^{3} - 5 a\) , \( 1\) , \( -5 a^{3} + 48 a^{2} + 9 a - 200\) , \( -205 a^{3} + 289 a^{2} + 857 a - 946\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-5a^{3}+48a^{2}+9a-200\right){x}-205a^{3}+289a^{2}+857a-946$ |
| 21.2-a2 |
21.2-a |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{6} \cdot 7^{2} \) |
$14.53358$ |
$(a), (a^2-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1211.072013$ |
3.631551357 |
\( \frac{90335681}{1323} a^{3} - \frac{28963072}{147} a^{2} + \frac{19006487}{441} a + \frac{132273482}{1323} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( a^{3} - 5 a\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 11 a^{3} + 3 a^{2} - 50 a - 24\) , \( -35 a^{3} - 4 a^{2} + 171 a + 91\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(11a^{3}+3a^{2}-50a-24\right){x}-35a^{3}-4a^{2}+171a+91$ |
| 21.2-a3 |
21.2-a |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{6} \) |
$14.53358$ |
$(a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$134.5635571$ |
3.631551357 |
\( -\frac{162147235}{352947} a^{3} + \frac{30715110}{117649} a^{2} + \frac{240844733}{117649} a - \frac{107377993}{352947} \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - a^{2} - 5 a + 1\) , \( a^{2} + a - 3\) , \( -9 a^{3} + 12 a^{2} + 37 a - 41\) , \( -55 a^{3} + 86 a^{2} + 226 a - 294\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+1\right){x}^{2}+\left(-9a^{3}+12a^{2}+37a-41\right){x}-55a^{3}+86a^{2}+226a-294$ |
| 21.2-a4 |
21.2-a |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{3} \cdot 7^{4} \) |
$14.53358$ |
$(a), (a^2-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$1211.072013$ |
3.631551357 |
\( -\frac{1790658835115600}{21609} a^{3} + \frac{5280430769506931}{21609} a^{2} - \frac{1340452888981817}{21609} a - \frac{306057095297029}{2401} \) |
\( \bigl[a\) , \( -a^{2} - a + 4\) , \( a^{2} - 2\) , \( -3 a^{3} - 4 a^{2} + 37 a - 32\) , \( -10 a^{3} + 38 a^{2} - 35 a + 5\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-3a^{3}-4a^{2}+37a-32\right){x}-10a^{3}+38a^{2}-35a+5$ |
| 21.2-b1 |
21.2-b |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{12} \) |
$14.53358$ |
$(a), (a^2-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$428.1363614$ |
1.283820587 |
\( \frac{22865829504156704}{41523861603} a^{3} - \frac{34444842533329259}{41523861603} a^{2} - \frac{93278703923001637}{41523861603} a + \frac{39651957308694504}{13841287201} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{3} - a^{2} - 5 a + 1\) , \( a\) , \( -11 a^{3} + 18 a^{2} + 45 a - 66\) , \( 22 a^{3} - 40 a^{2} - 88 a + 141\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a+1\right){x}^{2}+\left(-11a^{3}+18a^{2}+45a-66\right){x}+22a^{3}-40a^{2}-88a+141$ |
| 21.2-b2 |
21.2-b |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{6} \) |
$14.53358$ |
$(a), (a^2-2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$428.1363614$ |
1.283820587 |
\( -\frac{162147235}{352947} a^{3} + \frac{30715110}{117649} a^{2} + \frac{240844733}{117649} a - \frac{107377993}{352947} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{3} - a^{2} - 5 a + 1\) , \( a\) , \( -a^{3} - 2 a^{2} + 5 a + 9\) , \( 2 a^{3} - a^{2} - 9 a\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-a^{2}-5a+1\right){x}^{2}+\left(-a^{3}-2a^{2}+5a+9\right){x}+2a^{3}-a^{2}-9a$ |
| 21.2-b3 |
21.2-b |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{3} \cdot 7^{4} \) |
$14.53358$ |
$(a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$47.57070683$ |
1.283820587 |
\( -\frac{1790658835115600}{21609} a^{3} + \frac{5280430769506931}{21609} a^{2} - \frac{1340452888981817}{21609} a - \frac{306057095297029}{2401} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 3\) , \( a + 1\) , \( 0\) , \( -20 a^{3} + 20 a^{2} + 72 a - 84\) , \( -105 a^{3} + 88 a^{2} + 369 a - 393\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+3\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-20a^{3}+20a^{2}+72a-84\right){x}-105a^{3}+88a^{2}+369a-393$ |
| 21.2-b4 |
21.2-b |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
21.2 |
\( 3 \cdot 7 \) |
\( 3^{6} \cdot 7^{2} \) |
$14.53358$ |
$(a), (a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$47.57070683$ |
1.283820587 |
\( \frac{90335681}{1323} a^{3} - \frac{28963072}{147} a^{2} + \frac{19006487}{441} a + \frac{132273482}{1323} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 3\) , \( a + 1\) , \( 0\) , \( 5 a^{3} - 5 a^{2} - 18 a + 21\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+3\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5a^{3}-5a^{2}-18a+21\right){x}$ |
| 23.1-a1 |
23.1-a |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$14.69979$ |
$(a^3+a^2-6a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$785.9636754$ |
1.767608007 |
\( -\frac{37052}{23} a^{3} + \frac{149049}{23} a^{2} - \frac{94767}{23} a - \frac{37373}{23} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - 4 a - 1\) , \( a^{3} - a^{2} - 2 a + 3\) , \( a^{3} - a^{2} - 1\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(a^{3}-a^{2}-2a+3\right){x}+a^{3}-a^{2}-1$ |
| 23.1-a2 |
23.1-a |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( 23^{2} \) |
$14.69979$ |
$(a^3+a^2-6a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$392.9818377$ |
1.767608007 |
\( -\frac{50633045419}{529} a^{3} + \frac{150365991834}{529} a^{2} - \frac{38420556645}{529} a - \frac{78498997693}{529} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} - 4 a - 1\) , \( 6 a^{3} - 16 a^{2} + 3 a + 8\) , \( 30 a^{3} - 84 a^{2} + 14 a + 46\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(6a^{3}-16a^{2}+3a+8\right){x}+30a^{3}-84a^{2}+14a+46$ |
| 23.1-a3 |
23.1-a |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( 23^{6} \) |
$14.69979$ |
$(a^3+a^2-6a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$392.9818377$ |
1.767608007 |
\( \frac{486831020600336873949}{148035889} a^{3} - \frac{761271747411561789052}{148035889} a^{2} - \frac{2005004201343881350727}{148035889} a + \frac{2590771920905842207274}{148035889} \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( a^{3} - a^{2} - 4 a + 3\) , \( -8 a^{3} - 14 a^{2} + 6 a + 9\) , \( -73 a^{3} - 95 a^{2} + 140 a + 92\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(-8a^{3}-14a^{2}+6a+9\right){x}-73a^{3}-95a^{2}+140a+92$ |
| 23.1-a4 |
23.1-a |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( - 23^{3} \) |
$14.69979$ |
$(a^3+a^2-6a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$785.9636754$ |
1.767608007 |
\( \frac{10850010818}{12167} a^{3} - \frac{9511995091}{12167} a^{2} - \frac{39069480161}{12167} a + \frac{41191148260}{12167} \) |
\( \bigl[a^{3} - 3 a\) , \( a^{3} - 2 a^{2} - 3 a + 5\) , \( a^{3} - a^{2} - 4 a + 3\) , \( -8 a^{3} - 14 a^{2} + 11 a + 19\) , \( -81 a^{3} - 111 a^{2} + 147 a + 107\bigr] \) |
${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+5\right){x}^{2}+\left(-8a^{3}-14a^{2}+11a+19\right){x}-81a^{3}-111a^{2}+147a+107$ |
| 23.1-b1 |
23.1-b |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( 23^{2} \) |
$14.69979$ |
$(a^3+a^2-6a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.322577015$ |
$675.5727077$ |
3.920838811 |
\( -\frac{50633045419}{529} a^{3} + \frac{150365991834}{529} a^{2} - \frac{38420556645}{529} a - \frac{78498997693}{529} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 4 a - 1\) , \( 0\) , \( -4 a - 8\) , \( -14 a^{3} - 4 a^{2} + 51 a + 11\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(-4a-8\right){x}-14a^{3}-4a^{2}+51a+11$ |
| 23.1-b2 |
23.1-b |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( -23 \) |
$14.69979$ |
$(a^3+a^2-6a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$0.161288507$ |
$2702.290831$ |
3.920838811 |
\( -\frac{37052}{23} a^{3} + \frac{149049}{23} a^{2} - \frac{94767}{23} a - \frac{37373}{23} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 4 a - 1\) , \( 0\) , \( a + 2\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(a+2\right){x}$ |
| 23.1-b3 |
23.1-b |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( - 23^{3} \) |
$14.69979$ |
$(a^3+a^2-6a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.483865523$ |
$300.2545367$ |
3.920838811 |
\( \frac{10850010818}{12167} a^{3} - \frac{9511995091}{12167} a^{2} - \frac{39069480161}{12167} a + \frac{41191148260}{12167} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 3 a + 1\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 2 a^{3} - 17 a^{2} + 19 a + 14\) , \( 25 a^{3} - 77 a^{2} + 31 a + 42\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(2a^{3}-17a^{2}+19a+14\right){x}+25a^{3}-77a^{2}+31a+42$ |
| 23.1-b4 |
23.1-b |
$4$ |
$6$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
23.1 |
\( 23 \) |
\( 23^{6} \) |
$14.69979$ |
$(a^3+a^2-6a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.967731047$ |
$75.06363419$ |
3.920838811 |
\( \frac{486831020600336873949}{148035889} a^{3} - \frac{761271747411561789052}{148035889} a^{2} - \frac{2005004201343881350727}{148035889} a + \frac{2590771920905842207274}{148035889} \) |
\( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 3 a + 1\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 7 a^{3} - 32 a^{2} + 24 a + 19\) , \( -18 a^{3} + 51 a^{2} - 3 a - 28\bigr] \) |
${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(7a^{3}-32a^{2}+24a+19\right){x}-18a^{3}+51a^{2}-3a-28$ |
| 27.1-a1 |
27.1-a |
$1$ |
$1$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{26} \) |
$14.99739$ |
$(a), (a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$63.92801273$ |
2.300353987 |
\( -\frac{8805898963}{59049} a^{3} - \frac{1409689904}{59049} a^{2} + \frac{42396079244}{59049} a + \frac{7590005381}{19683} \) |
\( \bigl[a^{3} - 4 a\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{2} + a - 3\) , \( a^{3} - 3 a^{2} - 4 a + 6\) , \( -5 a^{3} + 16 a^{2} - 5 a - 17\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(a^{3}-3a^{2}-4a+6\right){x}-5a^{3}+16a^{2}-5a-17$ |
| 27.1-b1 |
27.1-b |
$2$ |
$2$ |
4.4.12357.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{16} \) |
$14.99739$ |
$(a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.133046076$ |
$607.7056968$ |
2.909369278 |
\( \frac{38802354404944}{81} a^{3} + \frac{52493491110833}{81} a^{2} - \frac{23500952042917}{27} a - \frac{16491691029511}{27} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 3\) , \( a^{3} - 5 a\) , \( 0\) , \( 27 a^{3} + 2 a^{2} - 128 a - 60\) , \( -170 a^{3} - 24 a^{2} + 817 a + 425\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+3\right){x}{y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(27a^{3}+2a^{2}-128a-60\right){x}-170a^{3}-24a^{2}+817a+425$ |
*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.
