| 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 |
| 48.1-a1 |
48.1-a |
$2$ |
$5$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{10} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$796.9802678$ |
2.264356376 |
\( \frac{53998}{243} a^{3} - \frac{174017}{243} a^{2} - \frac{198685}{243} a + \frac{1780081}{486} \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( \frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 5 a + 2\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( -\frac{8}{3} a^{3} + \frac{10}{3} a^{2} + 21 a - 21\) , \( -\frac{38}{3} a^{3} + \frac{52}{3} a^{2} + 96 a - 111\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){y}={x}^{3}+\left(\frac{2}{3}a^{3}-\frac{1}{3}a^{2}-5a+2\right){x}^{2}+\left(-\frac{8}{3}a^{3}+\frac{10}{3}a^{2}+21a-21\right){x}-\frac{38}{3}a^{3}+\frac{52}{3}a^{2}+96a-111$ |
| 48.1-a2 |
48.1-a |
$2$ |
$5$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{20} \cdot 3^{2} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \cdot 5 \) |
$1$ |
$1.275168428$ |
2.264356376 |
\( \frac{188222875363696875}{8} a^{3} - \frac{1522351303597490689}{48} a^{2} - \frac{8504923474410890561}{48} a + \frac{19471549236222137689}{96} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\) , \( \frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 4\) , \( 0\) , \( \frac{119}{3} a^{3} - \frac{142}{3} a^{2} - 150 a - 126\) , \( 2041 a^{3} - 6773 a^{2} + 2003 a + 1146\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a\right){x}{y}={x}^{3}+\left(\frac{1}{3}a^{3}-\frac{2}{3}a^{2}-2a+4\right){x}^{2}+\left(\frac{119}{3}a^{3}-\frac{142}{3}a^{2}-150a-126\right){x}+2041a^{3}-6773a^{2}+2003a+1146$ |
| 48.1-b1 |
48.1-b |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$32.82303819$ |
1.398837440 |
\( \frac{929865652521983315}{27} a^{3} - \frac{1138774984127055649}{9} a^{2} + \frac{1130881571253629065}{18} a + \frac{2086461174223332295}{54} \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 2\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 1\) , \( -\frac{82}{3} a^{3} + \frac{113}{3} a^{2} + 211 a - 238\) , \( \frac{478}{3} a^{3} - \frac{626}{3} a^{2} - 1188 a + 1349\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-1\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-2\right){x}^{2}+\left(-\frac{82}{3}a^{3}+\frac{113}{3}a^{2}+211a-238\right){x}+\frac{478}{3}a^{3}-\frac{626}{3}a^{2}-1188a+1349$ |
| 48.1-b2 |
48.1-b |
$2$ |
$3$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{12} \cdot 3^{2} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$295.4073437$ |
1.398837440 |
\( \frac{21224746042177}{24} a^{3} - \frac{9536989987379}{8} a^{2} - \frac{53280470046179}{8} a + \frac{182973377275723}{24} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 4 a - 2\) , \( a^{2} - a - 4\) , \( 11 a^{3} - 7 a^{2} - 67 a - 24\) , \( -\frac{121}{3} a^{3} - \frac{160}{3} a^{2} + 444 a + 159\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-4a-2\right){x}^{2}+\left(11a^{3}-7a^{2}-67a-24\right){x}-\frac{121}{3}a^{3}-\frac{160}{3}a^{2}+444a+159$ |
| 48.1-c1 |
48.1-c |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{12} \cdot 3^{4} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.046519900$ |
$1176.922004$ |
6.222195700 |
\( -\frac{2753009}{18} a^{3} - \frac{10565887}{36} a^{2} + \frac{26668097}{72} a + \frac{5985271}{36} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 1\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 4 a + 2\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 2\) , \( -3 a^{3} - a^{2} + 22 a + 6\) , \( -2 a^{3} + 18 a + 5\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-1\right){x}{y}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-2\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+4a+2\right){x}^{2}+\left(-3a^{3}-a^{2}+22a+6\right){x}-2a^{3}+18a+5$ |
| 48.1-d1 |
48.1-d |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{8} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.100071083$ |
$251.1895626$ |
5.713436273 |
\( \frac{900817}{162} a^{3} - \frac{1470923}{162} a^{2} - \frac{3168587}{81} a + \frac{8660297}{162} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\) , \( -a^{2} + a + 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 2\) , \( 2 a^{3} + a^{2} - 19 a - 7\) , \( \frac{25}{3} a^{3} + \frac{1}{3} a^{2} - 64 a - 24\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a\right){x}{y}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-2\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(2a^{3}+a^{2}-19a-7\right){x}+\frac{25}{3}a^{3}+\frac{1}{3}a^{2}-64a-24$ |
| 48.1-e1 |
48.1-e |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{16} \cdot 3 \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$125.7952267$ |
3.574056165 |
\( \frac{76926043}{48} a^{3} + \frac{1349417}{8} a^{2} - \frac{303233585}{24} a - \frac{8699767}{2} \) |
\( \bigl[a^{2} - a - 4\) , \( -\frac{2}{3} a^{3} + \frac{1}{3} a^{2} + 5 a - 2\) , \( a^{2} - a - 4\) , \( -\frac{2}{3} a^{3} + \frac{1}{3} a^{2} + 6 a - 5\) , \( \frac{7}{3} a^{3} - \frac{11}{3} a^{2} - 8 a - 4\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-\frac{2}{3}a^{3}+\frac{1}{3}a^{2}+5a-2\right){x}^{2}+\left(-\frac{2}{3}a^{3}+\frac{1}{3}a^{2}+6a-5\right){x}+\frac{7}{3}a^{3}-\frac{11}{3}a^{2}-8a-4$ |
| 48.1-f1 |
48.1-f |
$3$ |
$9$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$81$ |
\( 2 \) |
$1$ |
$2.732542865$ |
3.144265549 |
\( \frac{797557111969462491421}{3} a^{3} - \frac{1075105423836417293287}{3} a^{2} - \frac{4004214876876210459835}{2} a + \frac{13751062132018675906879}{6} \) |
\( \bigl[a\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a\) , \( a^{2} - 3\) , \( \frac{325}{3} a^{3} + \frac{112}{3} a^{2} - 865 a - 498\) , \( \frac{5150}{3} a^{3} + \frac{941}{3} a^{2} - 13566 a - 5705\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+2a\right){x}^{2}+\left(\frac{325}{3}a^{3}+\frac{112}{3}a^{2}-865a-498\right){x}+\frac{5150}{3}a^{3}+\frac{941}{3}a^{2}-13566a-5705$ |
| 48.1-f2 |
48.1-f |
$3$ |
$9$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{12} \cdot 3^{6} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$221.3359721$ |
3.144265549 |
\( \frac{280173925}{216} a^{3} - \frac{125912675}{72} a^{2} - \frac{703303475}{72} a + \frac{2415702223}{216} \) |
\( \bigl[a\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a\) , \( a^{2} - 3\) , \( \frac{5}{3} a^{3} + \frac{2}{3} a^{2} - 15 a - 8\) , \( \frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 6 a - 5\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+2a\right){x}^{2}+\left(\frac{5}{3}a^{3}+\frac{2}{3}a^{2}-15a-8\right){x}+\frac{2}{3}a^{3}-\frac{1}{3}a^{2}-6a-5$ |
| 48.1-f3 |
48.1-f |
$3$ |
$9$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1992.023749$ |
3.144265549 |
\( \frac{165192620}{3} a^{3} - \frac{1676553739}{6} a^{2} + \frac{895634596}{3} a + \frac{420165101}{3} \) |
\( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( \frac{4}{3} a^{3} + \frac{1}{3} a^{2} - 3 a + 1\) , \( \frac{13}{3} a^{3} + \frac{28}{3} a^{2} - 11 a - 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(\frac{4}{3}a^{3}+\frac{1}{3}a^{2}-3a+1\right){x}+\frac{13}{3}a^{3}+\frac{28}{3}a^{2}-11a-5$ |
| 48.1-g1 |
48.1-g |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.352560036$ |
$351.6405927$ |
7.044647918 |
\( \frac{185538109}{6} a^{3} - \frac{83396763}{2} a^{2} - \frac{465693059}{2} a + \frac{1599654271}{6} \) |
\( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( -a^{2} + a + 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 2\) , \( -\frac{7}{3} a^{3} + \frac{11}{3} a^{2} + 14 a - 18\) , \( -\frac{31}{3} a^{3} + \frac{41}{3} a^{2} + 79 a - 96\bigr] \) |
${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}{y}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-2\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-\frac{7}{3}a^{3}+\frac{11}{3}a^{2}+14a-18\right){x}-\frac{31}{3}a^{3}+\frac{41}{3}a^{2}+79a-96$ |
| 48.1-h1 |
48.1-h |
$1$ |
$1$ |
4.4.19821.1 |
$4$ |
$[4, 0]$ |
48.1 |
\( 2^{4} \cdot 3 \) |
\( 2^{12} \cdot 3^{2} \) |
$20.41063$ |
$(-1/3a^3-1/3a^2+3a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$146.4314743$ |
2.080183513 |
\( \frac{3428551}{24} a^{3} - \frac{4237589}{8} a^{2} + \frac{2169555}{8} a + \frac{499865}{3} \) |
\( \bigl[1\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 1\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\) , \( a^{3} - 6 a^{2} + 8 a + 1\) , \( 8 a^{3} - 30 a^{2} + 14 a + 9\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-1\right){x}^{2}+\left(a^{3}-6a^{2}+8a+1\right){x}+8a^{3}-30a^{2}+14a+9$ |
*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.
