| 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 |
| 10.1-a1 |
10.1-a |
$2$ |
$3$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{18} \cdot 5^{3} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.331766971$ |
$299.5258831$ |
2.490211922 |
\( -\frac{18143260469}{64000} a^{3} + \frac{12173702639}{32000} a^{2} + \frac{20688063403}{16000} a - \frac{102310022683}{64000} \) |
\( \bigl[a^{3} - 4 a + 1\) , \( -a\) , \( a^{2} - 3\) , \( -4 a^{3} - 4 a^{2} + 9 a + 1\) , \( -17 a^{3} - 22 a^{2} + 36 a + 16\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}-a{x}^{2}+\left(-4a^{3}-4a^{2}+9a+1\right){x}-17a^{3}-22a^{2}+36a+16$ |
| 10.1-a2 |
10.1-a |
$2$ |
$3$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{54} \cdot 5 \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \) |
$0.995300915$ |
$3.697850409$ |
2.490211922 |
\( -\frac{104421194283869401}{671088640} a^{3} - \frac{65592039235049719}{335544320} a^{2} + \frac{28265898435686641}{83886080} a + \frac{92555681506655613}{671088640} \) |
\( \bigl[a^{3} - 4 a + 1\) , \( -a\) , \( a^{2} - 3\) , \( -339 a^{3} - 434 a^{2} + 719 a + 306\) , \( -8637 a^{3} - 10869 a^{2} + 18676 a + 7678\bigr] \) |
${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}-a{x}^{2}+\left(-339a^{3}-434a^{2}+719a+306\right){x}-8637a^{3}-10869a^{2}+18676a+7678$ |
| 10.1-b1 |
10.1-b |
$8$ |
$12$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{12} \cdot 5^{3} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.369583550$ |
$716.1018848$ |
3.316089596 |
\( -\frac{57583081}{8000} a^{3} + \frac{92313143}{2000} a^{2} - \frac{128192731}{4000} a - \frac{120790917}{8000} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 3\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} - 4 a\) , \( -29 a^{3} - 35 a^{2} + 67 a + 28\) , \( -250 a^{3} - 312 a^{2} + 546 a + 222\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-3\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+2\right){x}^{2}+\left(-29a^{3}-35a^{2}+67a+28\right){x}-250a^{3}-312a^{2}+546a+222$ |
| 10.1-b2 |
10.1-b |
$8$ |
$12$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5^{9} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1.108750650$ |
$8.840764010$ |
3.316089596 |
\( \frac{170246460623168112451}{7812500} a^{3} + \frac{53462209509857054872}{1953125} a^{2} - \frac{184365542195919851399}{3906250} a - \frac{150882643732061508493}{7812500} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} + a^{2} - 3 a - 3\) , \( a^{2} + a - 3\) , \( 172 a^{3} - 551 a^{2} + 326 a + 110\) , \( 4341 a^{3} - 13428 a^{2} + 6464 a + 3927\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(172a^{3}-551a^{2}+326a+110\right){x}+4341a^{3}-13428a^{2}+6464a+3927$ |
| 10.1-b3 |
10.1-b |
$8$ |
$12$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{9} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$4.435002603$ |
$8.840764010$ |
3.316089596 |
\( -\frac{2732063681131532298413}{3906250} a^{3} + \frac{8441065838478510499631}{3906250} a^{2} - \frac{3978392047229931711101}{3906250} a - \frac{1307437773480485111158}{1953125} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - 2\) , \( a\) , \( 24 a^{3} + 15 a^{2} - 118 a - 98\) , \( 175 a^{3} + 12 a^{2} - 839 a - 228\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(24a^{3}+15a^{2}-118a-98\right){x}+175a^{3}+12a^{2}-839a-228$ |
| 10.1-b4 |
10.1-b |
$8$ |
$12$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{3} \cdot 5^{3} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1.478334201$ |
$716.1018848$ |
3.316089596 |
\( \frac{129742133341}{500} a^{3} - \frac{175986226817}{500} a^{2} - \frac{585999555143}{500} a + \frac{363945784931}{250} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - 2\) , \( a\) , \( -6 a^{3} + 10 a^{2} + 27 a - 38\) , \( 17 a^{3} - 22 a^{2} - 76 a + 92\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-6a^{3}+10a^{2}+27a-38\right){x}+17a^{3}-22a^{2}-76a+92$ |
| 10.1-b5 |
10.1-b |
$8$ |
$12$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{3} \cdot 5^{12} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.478334201$ |
$358.0509424$ |
3.316089596 |
\( -\frac{1943723921964277833}{976562500} a^{3} - \frac{369057274848591579}{976562500} a^{2} + \frac{9279632299215839059}{976562500} a + \frac{1633512853465750497}{488281250} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a^{2} - 3\) , \( -6 a^{3} - 2 a^{2} + 5 a + 5\) , \( 12 a^{3} + 9 a^{2} - 20 a - 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-6a^{3}-2a^{2}+5a+5\right){x}+12a^{3}+9a^{2}-20a-9$ |
| 10.1-b6 |
10.1-b |
$8$ |
$12$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{6} \cdot 5^{6} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.739167100$ |
$1432.203769$ |
3.316089596 |
\( -\frac{406878961}{31250} a^{3} - \frac{1453963347}{125000} a^{2} + \frac{7973027737}{125000} a + \frac{8375457067}{125000} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a^{2} - 3\) , \( -a^{3} - 2 a^{2} + 5 a + 5\) , \( -a^{3} + a^{2} + 2 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-a^{3}-2a^{2}+5a+5\right){x}-a^{3}+a^{2}+2a-1$ |
| 10.1-b7 |
10.1-b |
$8$ |
$12$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{36} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$4.435002603$ |
$4.420382005$ |
3.316089596 |
\( \frac{7856676960299175668001817761}{29103830456733703613281250} a^{3} - \frac{27631448782715195011362669607}{29103830456733703613281250} a^{2} + \frac{71561916707521519718706567697}{29103830456733703613281250} a + \frac{14685953958821655498210937026}{14551915228366851806640625} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a^{2} - 3\) , \( 4 a^{3} - 82 a^{2} + 60 a + 35\) , \( 73 a^{3} - 606 a^{2} + 424 a + 227\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(4a^{3}-82a^{2}+60a+35\right){x}+73a^{3}-606a^{2}+424a+227$ |
| 10.1-b8 |
10.1-b |
$8$ |
$12$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{18} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$2.217501301$ |
$17.68152802$ |
3.316089596 |
\( -\frac{15164333906399358528}{3814697265625} a^{3} + \frac{138181750293129166897}{7629394531250} a^{2} - \frac{81423039235754101037}{7629394531250} a - \frac{47447665542168389867}{7629394531250} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a^{2} - 3\) , \( 4 a^{3} - 82 a^{2} + 65 a + 35\) , \( 56 a^{3} - 593 a^{2} + 430 a + 227\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(4a^{3}-82a^{2}+65a+35\right){x}+56a^{3}-593a^{2}+430a+227$ |
| 10.1-c1 |
10.1-c |
$4$ |
$4$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.195098330$ |
$411.6522212$ |
1.509433839 |
\( -\frac{142152661}{20} a^{3} + \frac{110986533}{5} a^{2} - \frac{111025351}{10} a - \frac{124866677}{20} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( -a^{2} + 2\) , \( a^{3} - 3 a + 1\) , \( -4 a^{3} - 4 a^{2} + 13 a + 7\) , \( -a^{3} + a^{2} + 7 a\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-3\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-4a^{3}-4a^{2}+13a+7\right){x}-a^{3}+a^{2}+7a$ |
| 10.1-c2 |
10.1-c |
$4$ |
$4$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{4} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.195098330$ |
$823.3044425$ |
1.509433839 |
\( -\frac{37440829549}{1250} a^{3} - \frac{7119488387}{1250} a^{2} + \frac{178723471327}{1250} a + \frac{31462654066}{625} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 3\) , \( a^{3} - a^{2} - 5 a + 3\) , \( 0\) , \( 5 a^{3} + a^{2} - 17 a + 7\) , \( 4 a^{3} + a^{2} - 11 a + 7\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-3\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-5a+3\right){x}^{2}+\left(5a^{3}+a^{2}-17a+7\right){x}+4a^{3}+a^{2}-11a+7$ |
| 10.1-c3 |
10.1-c |
$4$ |
$4$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5 \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.195098330$ |
$823.3044425$ |
1.509433839 |
\( \frac{18832953773724794693}{10} a^{3} + \frac{23657763477810708539}{10} a^{2} - \frac{40788365420862504299}{10} a - \frac{8347238427245277262}{5} \) |
\( \bigl[a + 1\) , \( a^{2} - 4\) , \( a^{3} + a^{2} - 4 a - 3\) , \( 36 a^{3} - a^{2} - 165 a - 52\) , \( 138 a^{3} + 36 a^{2} - 677 a - 242\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}+a^{2}-4a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(36a^{3}-a^{2}-165a-52\right){x}+138a^{3}+36a^{2}-677a-242$ |
| 10.1-c4 |
10.1-c |
$4$ |
$4$ |
4.4.11324.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{2} \) |
$12.68056$ |
$(a^3-5a), (a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.097549165$ |
$1646.608885$ |
1.509433839 |
\( \frac{9860231737}{25} a^{3} + \frac{24772865687}{50} a^{2} - \frac{42710342177}{50} a - \frac{17481075207}{50} \) |
\( \bigl[a + 1\) , \( a^{2} - 4\) , \( a^{3} + a^{2} - 4 a - 3\) , \( 11 a^{3} - a^{2} - 50 a - 12\) , \( -35 a^{3} - 9 a^{2} + 171 a + 58\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}+a^{2}-4a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(11a^{3}-a^{2}-50a-12\right){x}-35a^{3}-9a^{2}+171a+58$ |
*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.
