| 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 |
| 71.1-a1 |
71.1-a |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( -71 \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$7524.719829$ |
2.24285 |
\( \frac{89431847697}{71} a^{5} - \frac{43715369081}{71} a^{4} - \frac{513221401647}{71} a^{3} + \frac{208176406233}{71} a^{2} + \frac{493457568669}{71} a - \frac{59179686201}{71} \) |
\( \bigl[a^{5} - 6 a^{3} + 8 a + 1\) , \( -a^{5} + 7 a^{3} - 10 a\) , \( 0\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 10 a + 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{5}-6a^{3}+8a+1\right){x}{y}={x}^{3}+\left(-a^{5}+7a^{3}-10a\right){x}^{2}+\left(-a^{5}-a^{4}+6a^{3}+3a^{2}-10a+1\right){x}$ |
| 71.1-a2 |
71.1-a |
$2$ |
$2$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( - 71^{2} \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$940.5899787$ |
2.24285 |
\( -\frac{11839647554558478959841}{5041} a^{5} + \frac{5787367720309411887888}{5041} a^{4} + \frac{67944035529605585955853}{5041} a^{3} - \frac{27559945908517340113981}{5041} a^{2} - \frac{65327540866640073976839}{5041} a + \frac{7834664827885457662413}{5041} \) |
\( \bigl[a^{5} - 6 a^{3} + 8 a + 1\) , \( -a^{5} + 7 a^{3} - 10 a\) , \( 0\) , \( 4 a^{5} + 4 a^{4} - 24 a^{3} - 12 a^{2} + 40 a - 4\) , \( 18 a^{5} + 27 a^{4} - 107 a^{3} - 80 a^{2} + 191 a - 21\bigr] \) |
${y}^2+\left(a^{5}-6a^{3}+8a+1\right){x}{y}={x}^{3}+\left(-a^{5}+7a^{3}-10a\right){x}^{2}+\left(4a^{5}+4a^{4}-24a^{3}-12a^{2}+40a-4\right){x}+18a^{5}+27a^{4}-107a^{3}-80a^{2}+191a-21$ |
| 71.1-b1 |
71.1-b |
$1$ |
$1$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( 71 \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.012503843$ |
$34628.08642$ |
3.09737 |
\( -\frac{2239709172}{71} a^{5} + \frac{1484198416}{71} a^{4} + \frac{13230268318}{71} a^{3} - \frac{6970381746}{71} a^{2} - \frac{14079872535}{71} a + \frac{1447849626}{71} \) |
\( \bigl[3 a^{5} - 2 a^{4} - 17 a^{3} + 11 a^{2} + 16 a - 4\) , \( -2 a^{5} + 2 a^{4} + 11 a^{3} - 11 a^{2} - 8 a + 5\) , \( a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 1\) , \( a^{5} - 6 a^{4} - 3 a^{3} + 36 a^{2} - 8 a - 37\) , \( -5 a^{5} + 11 a^{4} + 26 a^{3} - 60 a^{2} - 14 a + 49\bigr] \) |
${y}^2+\left(3a^{5}-2a^{4}-17a^{3}+11a^{2}+16a-4\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-1\right){y}={x}^{3}+\left(-2a^{5}+2a^{4}+11a^{3}-11a^{2}-8a+5\right){x}^{2}+\left(a^{5}-6a^{4}-3a^{3}+36a^{2}-8a-37\right){x}-5a^{5}+11a^{4}+26a^{3}-60a^{2}-14a+49$ |
| 71.1-c1 |
71.1-c |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( - 71^{9} \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$76.19196661$ |
1.83952 |
\( \frac{15074510965293950172967777821274}{45848500718449031} a^{5} - \frac{10246376051394215233897656562451}{45848500718449031} a^{4} - \frac{88900688178482615252690302298347}{45848500718449031} a^{3} + \frac{48445404031931088186318067724029}{45848500718449031} a^{2} + \frac{94110746850176071444600453149653}{45848500718449031} a - \frac{11417621517279237044933795673020}{45848500718449031} \) |
\( \bigl[a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 3 a - 2\) , \( -a^{5} + 7 a^{3} + a^{2} - 9 a - 4\) , \( a^{2} + a - 1\) , \( -15 a^{5} - 9 a^{4} + 70 a^{3} + 20 a^{2} - 72 a - 28\) , \( -111 a^{5} - 50 a^{4} + 481 a^{3} - 2 a^{2} - 394 a - 1\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+3a-2\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{5}+7a^{3}+a^{2}-9a-4\right){x}^{2}+\left(-15a^{5}-9a^{4}+70a^{3}+20a^{2}-72a-28\right){x}-111a^{5}-50a^{4}+481a^{3}-2a^{2}-394a-1$ |
| 71.1-c2 |
71.1-c |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( - 71^{3} \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$55543.94366$ |
1.83952 |
\( -\frac{29169505440}{357911} a^{5} + \frac{151117918400}{357911} a^{4} - \frac{25425543508}{357911} a^{3} - \frac{654677096029}{357911} a^{2} + \frac{727216594590}{357911} a - \frac{74279894665}{357911} \) |
\( \bigl[a + 1\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 9 a^{2} + 10 a - 2\) , \( a^{2} - 2\) , \( -a^{4} + 3 a^{2} + 1\) , \( -7 a^{5} + 5 a^{4} + 39 a^{3} - 20 a^{2} - 41 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(2a^{5}-2a^{4}-11a^{3}+9a^{2}+10a-2\right){x}^{2}+\left(-a^{4}+3a^{2}+1\right){x}-7a^{5}+5a^{4}+39a^{3}-20a^{2}-41a+4$ |
| 71.1-c3 |
71.1-c |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( - 71^{18} \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$324$ |
\( 2 \) |
$1$ |
$9.523995827$ |
1.83952 |
\( -\frac{32384932030645477671104879759286297815174}{2102085018129621311776010144838961} a^{5} + \frac{21844148113879633966889059724712269539776}{2102085018129621311776010144838961} a^{4} + \frac{190860263774172764626726183633974666350953}{2102085018129621311776010144838961} a^{3} - \frac{103295526582756225572298575973860225748056}{2102085018129621311776010144838961} a^{2} - \frac{201599340834596793007844079193785826495439}{2102085018129621311776010144838961} a + \frac{24448457916825108615003230495913550870705}{2102085018129621311776010144838961} \) |
\( \bigl[a + 1\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 9 a^{2} + 10 a - 2\) , \( a^{2} - 2\) , \( -165 a^{5} + 124 a^{4} + 1015 a^{3} - 622 a^{2} - 1135 a + 136\) , \( -2020 a^{5} + 1467 a^{4} + 12003 a^{3} - 6970 a^{2} - 13017 a + 1579\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(2a^{5}-2a^{4}-11a^{3}+9a^{2}+10a-2\right){x}^{2}+\left(-165a^{5}+124a^{4}+1015a^{3}-622a^{2}-1135a+136\right){x}-2020a^{5}+1467a^{4}+12003a^{3}-6970a^{2}-13017a+1579$ |
| 71.1-c4 |
71.1-c |
$4$ |
$6$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( - 71^{6} \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$6942.992958$ |
1.83952 |
\( \frac{66054493429974005697994}{128100283921} a^{5} - \frac{195272188967789233216758}{128100283921} a^{4} - \frac{143547654652558570361494}{128100283921} a^{3} + \frac{863864093776112345760668}{128100283921} a^{2} - \frac{693943093157691876933631}{128100283921} a + \frac{69078108932255278893563}{128100283921} \) |
\( \bigl[a + 1\) , \( 2 a^{5} - 2 a^{4} - 11 a^{3} + 9 a^{2} + 10 a - 2\) , \( a^{2} - 2\) , \( 10 a^{5} - 11 a^{4} - 45 a^{3} + 28 a^{2} + 45 a - 4\) , \( -16 a^{5} + 45 a^{4} - a^{3} - 49 a^{2} + 6 a - 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(2a^{5}-2a^{4}-11a^{3}+9a^{2}+10a-2\right){x}^{2}+\left(10a^{5}-11a^{4}-45a^{3}+28a^{2}+45a-4\right){x}-16a^{5}+45a^{4}-a^{3}-49a^{2}+6a-1$ |
| 71.1-d1 |
71.1-d |
$2$ |
$5$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( 71 \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$0.220956997$ |
$41839.53297$ |
2.64531 |
\( \frac{1061831}{71} a^{5} - \frac{19223}{71} a^{4} - \frac{5239532}{71} a^{3} + \frac{1358533}{71} a^{2} + \frac{4577495}{71} a - \frac{444281}{71} \) |
\( \bigl[2 a^{5} - 2 a^{4} - 11 a^{3} + 10 a^{2} + 9 a - 4\) , \( a^{5} - 5 a^{3} - a^{2} + 4 a + 2\) , \( 2 a^{5} - a^{4} - 12 a^{3} + 6 a^{2} + 14 a - 3\) , \( -4 a^{5} + 3 a^{4} + 22 a^{3} - 14 a^{2} - 21 a + 8\) , \( -2 a^{5} + 2 a^{4} + 11 a^{3} - 12 a^{2} - 8 a + 5\bigr] \) |
${y}^2+\left(2a^{5}-2a^{4}-11a^{3}+10a^{2}+9a-4\right){x}{y}+\left(2a^{5}-a^{4}-12a^{3}+6a^{2}+14a-3\right){y}={x}^{3}+\left(a^{5}-5a^{3}-a^{2}+4a+2\right){x}^{2}+\left(-4a^{5}+3a^{4}+22a^{3}-14a^{2}-21a+8\right){x}-2a^{5}+2a^{4}+11a^{3}-12a^{2}-8a+5$ |
| 71.1-d2 |
71.1-d |
$2$ |
$5$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( 71^{5} \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 5 \) |
$1.104784988$ |
$2.677730110$ |
2.64531 |
\( \frac{87005242380951413447981}{1804229351} a^{5} - \frac{358352300293503552738053}{1804229351} a^{4} + \frac{324582348954291382628292}{1804229351} a^{3} + \frac{267564005100390507857116}{1804229351} a^{2} - \frac{389861672438050834540672}{1804229351} a + \frac{41146371824560894280341}{1804229351} \) |
\( \bigl[a^{5} - a^{4} - 6 a^{3} + 5 a^{2} + 6 a - 1\) , \( -a^{5} + 5 a^{3} - a^{2} - 4 a + 2\) , \( a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 4 a - 1\) , \( -36 a^{5} - 27 a^{4} + 166 a^{3} + 15 a^{2} - 135 a - 7\) , \( -39 a^{5} - 341 a^{4} - 159 a^{3} + 650 a^{2} + 393 a - 108\bigr] \) |
${y}^2+\left(a^{5}-a^{4}-6a^{3}+5a^{2}+6a-1\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+4a-1\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-4a+2\right){x}^{2}+\left(-36a^{5}-27a^{4}+166a^{3}+15a^{2}-135a-7\right){x}-39a^{5}-341a^{4}-159a^{3}+650a^{2}+393a-108$ |
| 71.1-e1 |
71.1-e |
$1$ |
$1$ |
6.6.703493.1 |
$6$ |
$[6, 0]$ |
71.1 |
\( 71 \) |
\( 71^{5} \) |
$106.91533$ |
$(-2a^5+a^4+12a^3-4a^2-12a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$866.1300830$ |
1.03265 |
\( -\frac{124315078720314611694730}{1804229351} a^{5} + \frac{84498863776551023357538}{1804229351} a^{4} + \frac{733137948754350072437846}{1804229351} a^{3} - \frac{399515065412184994079968}{1804229351} a^{2} - \frac{776103776934549949304354}{1804229351} a + \frac{94157783375202703558381}{1804229351} \) |
\( \bigl[3 a^{5} - 2 a^{4} - 17 a^{3} + 11 a^{2} + 16 a - 5\) , \( -a^{5} + 7 a^{3} + a^{2} - 9 a - 3\) , \( a^{5} - 6 a^{3} + a^{2} + 7 a\) , \( -69 a^{5} + 47 a^{4} + 407 a^{3} - 221 a^{2} - 427 a + 54\) , \( -236 a^{5} + 160 a^{4} + 1392 a^{3} - 756 a^{2} - 1470 a + 178\bigr] \) |
${y}^2+\left(3a^{5}-2a^{4}-17a^{3}+11a^{2}+16a-5\right){x}{y}+\left(a^{5}-6a^{3}+a^{2}+7a\right){y}={x}^{3}+\left(-a^{5}+7a^{3}+a^{2}-9a-3\right){x}^{2}+\left(-69a^{5}+47a^{4}+407a^{3}-221a^{2}-427a+54\right){x}-236a^{5}+160a^{4}+1392a^{3}-756a^{2}-1470a+178$ |
*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.
