| 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 |
| 45.1-a1 |
45.1-a |
$2$ |
$2$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{3} \cdot 5 \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.149447375$ |
$1827.780293$ |
2.66508936 |
\( \frac{34992}{5} a^{4} - \frac{81598}{5} a^{3} - \frac{54171}{5} a^{2} + \frac{89407}{5} a + \frac{46644}{5} \) |
\( \bigl[-a^{4} + a^{3} + 5 a^{2} - 2 a - 4\) , \( 2 a^{4} - 3 a^{3} - 8 a^{2} + 9 a + 5\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 6\) , \( -2 a^{4} + 5 a^{3} + 4 a^{2} - 9 a + 1\) , \( 2 a^{4} - 2 a^{3} - 11 a^{2} + 10 a + 5\bigr] \) |
${y}^2+\left(-a^{4}+a^{3}+5a^{2}-2a-4\right){x}{y}+\left(-2a^{4}+3a^{3}+9a^{2}-9a-6\right){y}={x}^{3}+\left(2a^{4}-3a^{3}-8a^{2}+9a+5\right){x}^{2}+\left(-2a^{4}+5a^{3}+4a^{2}-9a+1\right){x}+2a^{4}-2a^{3}-11a^{2}+10a+5$ |
| 45.1-a2 |
45.1-a |
$2$ |
$2$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{3} \cdot 5^{2} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.298894751$ |
$456.9450733$ |
2.66508936 |
\( \frac{11314724189}{25} a^{4} - \frac{30736776391}{25} a^{3} - \frac{3562664832}{25} a^{2} + \frac{29192288229}{25} a + \frac{6597394298}{25} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 6\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 3\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 5\) , \( 16 a^{4} - 20 a^{3} - 76 a^{2} + 58 a + 56\) , \( -11 a^{4} + 10 a^{3} + 56 a^{2} - 13 a - 77\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+9a^{2}-9a-6\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-6a-5\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+4a+3\right){x}^{2}+\left(16a^{4}-20a^{3}-76a^{2}+58a+56\right){x}-11a^{4}+10a^{3}+56a^{2}-13a-77$ |
| 45.1-b1 |
45.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{6} \cdot 5^{3} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.2, 5B.4.2 |
$225$ |
\( 1 \) |
$1$ |
$1.656052230$ |
1.45417284 |
\( \frac{9227278873526453906244312610219036327836}{125} a^{4} + \frac{3459009377990835094505717267240191015816}{125} a^{3} - \frac{41380713886380859304809564621687978514293}{125} a^{2} - \frac{38438451731868504360194666541225269140904}{125} a - \frac{6711393728551335750914883813930912798473}{125} \) |
\( \bigl[-a^{4} + a^{3} + 5 a^{2} - 3 a - 3\) , \( -a^{4} + 5 a^{2} - 2\) , \( a^{3} - 4 a - 1\) , \( -527 a^{4} + 1330 a^{3} + 662 a^{2} - 1670 a - 937\) , \( 18278 a^{4} - 41440 a^{3} - 28524 a^{2} + 47105 a + 27660\bigr] \) |
${y}^2+\left(-a^{4}+a^{3}+5a^{2}-3a-3\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{4}+5a^{2}-2\right){x}^{2}+\left(-527a^{4}+1330a^{3}+662a^{2}-1670a-937\right){x}+18278a^{4}-41440a^{3}-28524a^{2}+47105a+27660$ |
| 45.1-b2 |
45.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{6} \cdot 5 \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.1, 5B.4.2 |
$25$ |
\( 1 \) |
$1$ |
$134.1402306$ |
1.45417284 |
\( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \) |
\( \bigl[a\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 4\) , \( 330 a^{4} + 154 a^{3} - 1529 a^{2} - 1577 a - 314\) , \( 9329 a^{4} + 3425 a^{3} - 41530 a^{2} - 38029 a - 6558\bigr] \) |
${y}^2+a{x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-7a-4\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+7a+3\right){x}^{2}+\left(330a^{4}+154a^{3}-1529a^{2}-1577a-314\right){x}+9329a^{4}+3425a^{3}-41530a^{2}-38029a-6558$ |
| 45.1-b3 |
45.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{6} \cdot 5^{15} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.2, 5B.4.1 |
$9$ |
\( 1 \) |
$1$ |
$41.40130575$ |
1.45417284 |
\( \frac{175802229236840746}{30517578125} a^{4} + \frac{65933855205252401}{30517578125} a^{3} - \frac{788398234528339998}{30517578125} a^{2} - \frac{732499454042744644}{30517578125} a - \frac{127944679222457853}{30517578125} \) |
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 4 a\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 4\) , \( 9 a^{3} - 3 a^{2} - 40 a - 8\) , \( 10 a^{4} + 2 a^{3} - 44 a^{2} - 35 a - 7\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-7a-4\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+4a\right){x}^{2}+\left(9a^{3}-3a^{2}-40a-8\right){x}+10a^{4}+2a^{3}-44a^{2}-35a-7$ |
| 45.1-b4 |
45.1-b |
$4$ |
$15$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{6} \cdot 5^{5} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3, 5$ |
3B.1.1, 5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$3353.505766$ |
1.45417284 |
\( \frac{81829946}{3125} a^{4} - \frac{143808649}{3125} a^{3} - \frac{300612723}{3125} a^{2} + \frac{393132181}{3125} a + \frac{109010922}{3125} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{2} + a + 2\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 6 a - 3\) , \( 4 a^{4} - 5 a^{3} - 19 a^{2} + 13 a + 17\) , \( 8 a^{4} - 10 a^{3} - 37 a^{2} + 25 a + 32\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(-a^{4}+2a^{3}+4a^{2}-6a-3\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(4a^{4}-5a^{3}-19a^{2}+13a+17\right){x}+8a^{4}-10a^{3}-37a^{2}+25a+32$ |
| 45.1-c1 |
45.1-c |
$2$ |
$2$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{9} \cdot 5 \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$887.0117130$ |
1.73084764 |
\( \frac{34992}{5} a^{4} - \frac{81598}{5} a^{3} - \frac{54171}{5} a^{2} + \frac{89407}{5} a + \frac{46644}{5} \) |
\( \bigl[-a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 5\) , \( a^{4} - 5 a^{2} - 2 a + 2\) , \( a^{3} - 3 a\) , \( 2 a^{4} - 4 a^{3} - 11 a^{2} + 13 a + 13\) , \( 2 a^{4} - 3 a^{3} - 9 a^{2} + 5 a + 6\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+5a^{2}-6a-5\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{4}-5a^{2}-2a+2\right){x}^{2}+\left(2a^{4}-4a^{3}-11a^{2}+13a+13\right){x}+2a^{4}-3a^{3}-9a^{2}+5a+6$ |
| 45.1-c2 |
45.1-c |
$2$ |
$2$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{9} \cdot 5^{2} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$443.5058565$ |
1.73084764 |
\( \frac{11314724189}{25} a^{4} - \frac{30736776391}{25} a^{3} - \frac{3562664832}{25} a^{2} + \frac{29192288229}{25} a + \frac{6597394298}{25} \) |
\( \bigl[-a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 5\) , \( -3 a^{4} + 4 a^{3} + 13 a^{2} - 11 a - 9\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 4\) , \( 24 a^{4} - 31 a^{3} - 110 a^{2} + 79 a + 89\) , \( 93 a^{4} - 125 a^{3} - 415 a^{2} + 318 a + 327\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+5a^{2}-7a-5\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-6a-4\right){y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-11a-9\right){x}^{2}+\left(24a^{4}-31a^{3}-110a^{2}+79a+89\right){x}+93a^{4}-125a^{3}-415a^{2}+318a+327$ |
| 45.1-d1 |
45.1-d |
$1$ |
$1$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{6} \cdot 5 \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 1 \) |
$1$ |
$325.0490427$ |
1.26855229 |
\( \frac{144184611}{5} a^{4} - \frac{255843129}{5} a^{3} - \frac{524006273}{5} a^{2} + \frac{693618211}{5} a + \frac{186593617}{5} \) |
\( \bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 7\) , \( -3 a^{4} + 4 a^{3} + 13 a^{2} - 12 a - 9\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 3\) , \( -7 a^{4} + 2 a^{3} + 20 a^{2} - 11 a - 2\) , \( -11 a^{4} - 8 a^{3} + 24 a^{2} + 9 a + 2\bigr] \) |
${y}^2+\left(-2a^{4}+3a^{3}+9a^{2}-8a-7\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+3\right){y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-12a-9\right){x}^{2}+\left(-7a^{4}+2a^{3}+20a^{2}-11a-2\right){x}-11a^{4}-8a^{3}+24a^{2}+9a+2$ |
| 45.1-e1 |
45.1-e |
$4$ |
$4$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{10} \cdot 5^{7} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$0.062194931$ |
$365.0275762$ |
3.10104590 |
\( \frac{43762711342084459813388}{6328125} a^{4} + \frac{19657683179991310734076}{2109375} a^{3} - \frac{80370595538065362656194}{6328125} a^{2} - \frac{11238856761605347645073}{703125} a - \frac{18641764309297167053009}{6328125} \) |
\( \bigl[-a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 5\) , \( 3 a^{4} - 4 a^{3} - 13 a^{2} + 11 a + 9\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 5\) , \( -14 a^{4} + 10 a^{3} - 3 a^{2} + 2 a - 7\) , \( -284 a^{4} + 587 a^{3} + 1227 a^{2} - 1161 a - 1194\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+5a^{2}-7a-5\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-6a-5\right){y}={x}^{3}+\left(3a^{4}-4a^{3}-13a^{2}+11a+9\right){x}^{2}+\left(-14a^{4}+10a^{3}-3a^{2}+2a-7\right){x}-284a^{4}+587a^{3}+1227a^{2}-1161a-1194$ |
| 45.1-e2 |
45.1-e |
$4$ |
$4$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{14} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 7 \) |
$0.031097465$ |
$1460.110304$ |
3.10104590 |
\( \frac{810387038438023516}{54931640625} a^{4} + \frac{363395431073753132}{18310546875} a^{3} - \frac{1491237718490098883}{54931640625} a^{2} - \frac{207634903981355686}{6103515625} a - \frac{338342690307372463}{54931640625} \) |
\( \bigl[-a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 5\) , \( 3 a^{4} - 4 a^{3} - 13 a^{2} + 11 a + 9\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 5\) , \( -24 a^{4} + 30 a^{3} + 107 a^{2} - 73 a - 97\) , \( -172 a^{4} + 214 a^{3} + 809 a^{2} - 536 a - 735\bigr] \) |
${y}^2+\left(-a^{4}+2a^{3}+5a^{2}-7a-5\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-6a-5\right){y}={x}^{3}+\left(3a^{4}-4a^{3}-13a^{2}+11a+9\right){x}^{2}+\left(-24a^{4}+30a^{3}+107a^{2}-73a-97\right){x}-172a^{4}+214a^{3}+809a^{2}-536a-735$ |
| 45.1-e3 |
45.1-e |
$4$ |
$4$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{7} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$0.015548732$ |
$5840.441219$ |
3.10104590 |
\( \frac{1404373231}{234375} a^{4} - \frac{1413777188}{78125} a^{3} - \frac{830144678}{234375} a^{2} + \frac{1478965772}{78125} a + \frac{1473768167}{234375} \) |
\( \bigl[a^{2} - a - 1\) , \( 2 a^{4} - 2 a^{3} - 9 a^{2} + 6 a + 5\) , \( a^{3} - 4 a - 1\) , \( a^{3} - a^{2} - 2 a + 3\) , \( -2 a^{4} - a^{3} + 9 a^{2} + 10 a + 3\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(2a^{4}-2a^{3}-9a^{2}+6a+5\right){x}^{2}+\left(a^{3}-a^{2}-2a+3\right){x}-2a^{4}-a^{3}+9a^{2}+10a+3$ |
| 45.1-e4 |
45.1-e |
$4$ |
$4$ |
5.5.65657.1 |
$5$ |
$[5, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{7} \cdot 5^{28} \) |
$33.50428$ |
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \cdot 7 \) |
$0.062194931$ |
$45.62844702$ |
3.10104590 |
\( \frac{359045212142585076290656747468}{111758708953857421875} a^{4} - \frac{147682742439513102209920029764}{37252902984619140625} a^{3} - \frac{1691569060032060442379194043234}{111758708953857421875} a^{2} + \frac{371284744650378795256836683741}{37252902984619140625} a + \frac{1534625546447161055611719609551}{111758708953857421875} \) |
\( \bigl[a^{2} - a - 1\) , \( 2 a^{4} - 2 a^{3} - 9 a^{2} + 6 a + 5\) , \( a^{3} - 4 a - 1\) , \( 40 a^{4} - 14 a^{3} - 166 a^{2} - 52 a - 17\) , \( 84 a^{4} + 35 a^{3} - 366 a^{2} - 370 a - 111\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(2a^{4}-2a^{3}-9a^{2}+6a+5\right){x}^{2}+\left(40a^{4}-14a^{3}-166a^{2}-52a-17\right){x}+84a^{4}+35a^{3}-366a^{2}-370a-111$ |
*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.
