| 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 |
| 79.1-a1 |
79.1-a |
$1$ |
$1$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( 79 \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$1151.176491$ |
1.65278 |
\( -\frac{34707474083885637}{79} a^{5} - \frac{4278503312080601}{79} a^{4} + \frac{129745464680528828}{79} a^{3} - \frac{2174717524050824}{79} a^{2} - \frac{74032467724509348}{79} a + \frac{16346211679544252}{79} \) |
\( \bigl[a^{5} - 5 a^{3} + 5 a + 1\) , \( 3 a^{5} - 3 a^{4} - 14 a^{3} + 10 a^{2} + 13 a - 2\) , \( a^{5} - a^{4} - 4 a^{3} + 3 a^{2} + 2 a\) , \( 14 a^{5} - 9 a^{4} - 62 a^{3} + 35 a^{2} + 54 a - 11\) , \( 23 a^{5} - 12 a^{4} - 97 a^{3} + 48 a^{2} + 79 a - 22\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+5a+1\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+3a^{2}+2a\right){y}={x}^{3}+\left(3a^{5}-3a^{4}-14a^{3}+10a^{2}+13a-2\right){x}^{2}+\left(14a^{5}-9a^{4}-62a^{3}+35a^{2}+54a-11\right){x}+23a^{5}-12a^{4}-97a^{3}+48a^{2}+79a-22$ |
| 79.1-b1 |
79.1-b |
$4$ |
$4$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( - 79^{4} \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$135.6371329$ |
1.55791 |
\( \frac{12330755432171635825082498}{38950081} a^{5} - \frac{16242907791157107631688511}{38950081} a^{4} - \frac{60412576977068082076256075}{38950081} a^{3} + \frac{57400435367990039278537657}{38950081} a^{2} + \frac{63850633148601212149915983}{38950081} a - \frac{18060897650216400542227732}{38950081} \) |
\( \bigl[a^{4} - 4 a^{2} + 2\) , \( -a^{4} + a^{3} + 4 a^{2} - 3 a - 4\) , \( -a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 4 a\) , \( 7 a^{5} - 18 a^{4} - 4 a^{3} + 28 a^{2} - 3 a - 24\) , \( 654 a^{5} - 2509 a^{4} + 1909 a^{3} + 1882 a^{2} - 2039 a + 340\bigr] \) |
${y}^2+\left(a^{4}-4a^{2}+2\right){x}{y}+\left(-a^{5}+a^{4}+5a^{3}-3a^{2}-4a\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-3a-4\right){x}^{2}+\left(7a^{5}-18a^{4}-4a^{3}+28a^{2}-3a-24\right){x}+654a^{5}-2509a^{4}+1909a^{3}+1882a^{2}-2039a+340$ |
| 79.1-b2 |
79.1-b |
$4$ |
$4$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( -79 \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$17361.55302$ |
1.55791 |
\( \frac{1208028}{79} a^{5} + \frac{549606}{79} a^{4} - \frac{5168743}{79} a^{3} - \frac{3494134}{79} a^{2} + \frac{881969}{79} a + \frac{64612}{79} \) |
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 2 a - 3\) , \( -a^{5} + a^{4} + 5 a^{3} - 4 a^{2} - 6 a + 3\) , \( a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 2 a - 2\) , \( -3 a^{5} + 4 a^{4} + 15 a^{3} - 16 a^{2} - 18 a + 6\) , \( -4 a^{5} + 4 a^{4} + 19 a^{3} - 14 a^{2} - 18 a + 5\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-3\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-2\right){y}={x}^{3}+\left(-a^{5}+a^{4}+5a^{3}-4a^{2}-6a+3\right){x}^{2}+\left(-3a^{5}+4a^{4}+15a^{3}-16a^{2}-18a+6\right){x}-4a^{5}+4a^{4}+19a^{3}-14a^{2}-18a+5$ |
| 79.1-b3 |
79.1-b |
$4$ |
$4$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( 79^{2} \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$8680.776510$ |
1.55791 |
\( -\frac{24046309658430}{6241} a^{5} + \frac{5353689404761}{6241} a^{4} + \frac{108122257802003}{6241} a^{3} + \frac{685210524510}{6241} a^{2} - \frac{57231648199249}{6241} a + \frac{12349930820557}{6241} \) |
\( \bigl[a^{5} - 5 a^{3} + 4 a\) , \( 2 a^{5} - 3 a^{4} - 9 a^{3} + 11 a^{2} + 8 a - 5\) , \( a^{4} - a^{3} - 3 a^{2} + 3 a + 2\) , \( -53 a^{5} - 4 a^{4} + 204 a^{3} - 7 a^{2} - 127 a + 15\) , \( -545 a^{5} - 45 a^{4} + 2059 a^{3} - 102 a^{2} - 1220 a + 263\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+4a\right){x}{y}+\left(a^{4}-a^{3}-3a^{2}+3a+2\right){y}={x}^{3}+\left(2a^{5}-3a^{4}-9a^{3}+11a^{2}+8a-5\right){x}^{2}+\left(-53a^{5}-4a^{4}+204a^{3}-7a^{2}-127a+15\right){x}-545a^{5}-45a^{4}+2059a^{3}-102a^{2}-1220a+263$ |
| 79.1-b4 |
79.1-b |
$4$ |
$4$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( 79 \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$4340.388255$ |
1.55791 |
\( -\frac{8074169398297883371934}{79} a^{5} + \frac{2539553867757618298801}{79} a^{4} + \frac{36577024154073433239653}{79} a^{3} - \frac{2943812121021753307351}{79} a^{2} - \frac{21110051488039191505569}{79} a + \frac{4790450604778895020892}{79} \) |
\( \bigl[a^{5} - 5 a^{3} + 4 a\) , \( 2 a^{5} - 3 a^{4} - 9 a^{3} + 11 a^{2} + 8 a - 5\) , \( a^{4} - a^{3} - 3 a^{2} + 3 a + 2\) , \( -173 a^{5} - 19 a^{4} + 639 a^{3} - 12 a^{2} - 347 a + 50\) , \( 2533 a^{5} + 346 a^{4} - 9413 a^{3} + 37 a^{2} + 5248 a - 1108\bigr] \) |
${y}^2+\left(a^{5}-5a^{3}+4a\right){x}{y}+\left(a^{4}-a^{3}-3a^{2}+3a+2\right){y}={x}^{3}+\left(2a^{5}-3a^{4}-9a^{3}+11a^{2}+8a-5\right){x}^{2}+\left(-173a^{5}-19a^{4}+639a^{3}-12a^{2}-347a+50\right){x}+2533a^{5}+346a^{4}-9413a^{3}+37a^{2}+5248a-1108$ |
| 79.1-c1 |
79.1-c |
$1$ |
$1$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( 79^{2} \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.001818303$ |
$88821.88007$ |
2.78254 |
\( -\frac{29770178860}{6241} a^{5} + \frac{39707625788}{6241} a^{4} + \frac{144718022792}{6241} a^{3} - \frac{139902102717}{6241} a^{2} - \frac{150477209176}{6241} a + \frac{42622095976}{6241} \) |
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 3 a - 2\) , \( a^{5} - 2 a^{4} - 3 a^{3} + 6 a^{2} - 3\) , \( -a^{5} + 2 a^{4} + 5 a^{3} - 7 a^{2} - 6 a + 3\) , \( 3 a^{4} - 2 a^{3} - 7 a^{2} + 3 a + 2\) , \( 2 a^{5} - 6 a^{3} + 2 a - 1\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+3a-2\right){x}{y}+\left(-a^{5}+2a^{4}+5a^{3}-7a^{2}-6a+3\right){y}={x}^{3}+\left(a^{5}-2a^{4}-3a^{3}+6a^{2}-3\right){x}^{2}+\left(3a^{4}-2a^{3}-7a^{2}+3a+2\right){x}+2a^{5}-6a^{3}+2a-1$ |
| 79.1-d1 |
79.1-d |
$2$ |
$3$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( 79^{6} \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$678.6486118$ |
1.94872 |
\( \frac{44453593451476441317}{243087455521} a^{5} - \frac{78067113014900737157}{243087455521} a^{4} - \frac{196851531638275063103}{243087455521} a^{3} + \frac{307626136839531014225}{243087455521} a^{2} + \frac{163923774654485334193}{243087455521} a - \frac{182294605814822846245}{243087455521} \) |
\( \bigl[a^{4} - 3 a^{2} + 1\) , \( 2 a^{5} - 3 a^{4} - 8 a^{3} + 11 a^{2} + 4 a - 6\) , \( -a^{5} + 2 a^{4} + 4 a^{3} - 6 a^{2} - 3 a + 1\) , \( 6 a^{5} - a^{4} - 24 a^{3} + 6 a^{2} + 19 a - 4\) , \( 16 a^{5} - 9 a^{4} - 59 a^{3} + 34 a^{2} + 28 a - 8\bigr] \) |
${y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(-a^{5}+2a^{4}+4a^{3}-6a^{2}-3a+1\right){y}={x}^{3}+\left(2a^{5}-3a^{4}-8a^{3}+11a^{2}+4a-6\right){x}^{2}+\left(6a^{5}-a^{4}-24a^{3}+6a^{2}+19a-4\right){x}+16a^{5}-9a^{4}-59a^{3}+34a^{2}+28a-8$ |
| 79.1-d2 |
79.1-d |
$2$ |
$3$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( 79^{2} \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$678.6486118$ |
1.94872 |
\( -\frac{41788037337331}{6241} a^{5} - \frac{5151356449741}{6241} a^{4} + \frac{156214415866091}{6241} a^{3} - \frac{2618366872600}{6241} a^{2} - \frac{89135591080143}{6241} a + \frac{19680947110096}{6241} \) |
\( \bigl[a^{4} - a^{3} - 3 a^{2} + 2 a + 1\) , \( a + 1\) , \( -2 a^{5} + 3 a^{4} + 9 a^{3} - 10 a^{2} - 7 a + 3\) , \( 2 a^{5} - 2 a^{4} - 10 a^{3} + 8 a^{2} + 10 a - 5\) , \( a^{5} - 2 a^{4} - 4 a^{3} + 8 a^{2} + 3 a - 7\bigr] \) |
${y}^2+\left(a^{4}-a^{3}-3a^{2}+2a+1\right){x}{y}+\left(-2a^{5}+3a^{4}+9a^{3}-10a^{2}-7a+3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a^{5}-2a^{4}-10a^{3}+8a^{2}+10a-5\right){x}+a^{5}-2a^{4}-4a^{3}+8a^{2}+3a-7$ |
| 79.1-e1 |
79.1-e |
$1$ |
$1$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( -79 \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.003964302$ |
$73613.05735$ |
2.51389 |
\( \frac{442196209664}{79} a^{5} - \frac{139077779456}{79} a^{4} - \frac{2003217559552}{79} a^{3} + \frac{161219244032}{79} a^{2} + \frac{1156138635264}{79} a - \frac{262359261184}{79} \) |
\( \bigl[0\) , \( -2 a^{5} + 3 a^{4} + 9 a^{3} - 11 a^{2} - 9 a + 4\) , \( -a^{5} + 2 a^{4} + 4 a^{3} - 6 a^{2} - 2 a + 1\) , \( a^{5} - a^{4} - 7 a^{3} + 3 a^{2} + 12 a + 1\) , \( -2 a^{5} + 4 a^{4} + 12 a^{3} - 15 a^{2} - 17 a + 7\bigr] \) |
${y}^2+\left(-a^{5}+2a^{4}+4a^{3}-6a^{2}-2a+1\right){y}={x}^{3}+\left(-2a^{5}+3a^{4}+9a^{3}-11a^{2}-9a+4\right){x}^{2}+\left(a^{5}-a^{4}-7a^{3}+3a^{2}+12a+1\right){x}-2a^{5}+4a^{4}+12a^{3}-15a^{2}-17a+7$ |
| 79.1-f1 |
79.1-f |
$2$ |
$3$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( 79 \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$685.1893380$ |
0.983747 |
\( \frac{1025326236369}{79} a^{5} - \frac{1798397134160}{79} a^{4} - \frac{4560527408010}{79} a^{3} + \frac{7044300358011}{79} a^{2} + \frac{3783035053814}{79} a - \frac{4176336847502}{79} \) |
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 4 a^{2} + 2 a - 3\) , \( -2 a^{5} + 3 a^{4} + 10 a^{3} - 11 a^{2} - 12 a + 3\) , \( a^{4} - 4 a^{2} + 3\) , \( 6 a^{5} - 12 a^{4} - 32 a^{3} + 43 a^{2} + 38 a - 16\) , \( -4 a^{5} + 5 a^{4} + 25 a^{3} - 14 a^{2} - 37 a - 4\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-3\right){x}{y}+\left(a^{4}-4a^{2}+3\right){y}={x}^{3}+\left(-2a^{5}+3a^{4}+10a^{3}-11a^{2}-12a+3\right){x}^{2}+\left(6a^{5}-12a^{4}-32a^{3}+43a^{2}+38a-16\right){x}-4a^{5}+5a^{4}+25a^{3}-14a^{2}-37a-4$ |
| 79.1-f2 |
79.1-f |
$2$ |
$3$ |
6.6.485125.1 |
$6$ |
$[6, 0]$ |
79.1 |
\( 79 \) |
\( 79^{3} \) |
$89.57792$ |
$(a^4+a^3-3a^2-3a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$685.1893380$ |
0.983747 |
\( -\frac{239686761033}{493039} a^{5} + \frac{315209365865}{493039} a^{4} + \frac{1175549500213}{493039} a^{3} - \frac{1115074985589}{493039} a^{2} - \frac{1242991778332}{493039} a + \frac{350164433016}{493039} \) |
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 3 a^{2} + 3 a - 1\) , \( -a^{5} + 2 a^{4} + 4 a^{3} - 8 a^{2} - 3 a + 6\) , \( -a^{5} + 2 a^{4} + 4 a^{3} - 6 a^{2} - 3 a + 2\) , \( -2 a^{5} + 2 a^{4} + 9 a^{3} - 8 a^{2} - 12 a + 11\) , \( -7 a^{5} + 19 a^{4} + 13 a^{3} - 62 a^{2} + 33 a - 1\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-4a^{3}+3a^{2}+3a-1\right){x}{y}+\left(-a^{5}+2a^{4}+4a^{3}-6a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+4a^{3}-8a^{2}-3a+6\right){x}^{2}+\left(-2a^{5}+2a^{4}+9a^{3}-8a^{2}-12a+11\right){x}-7a^{5}+19a^{4}+13a^{3}-62a^{2}+33a-1$ |
*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.
