| 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 |
| 21.1-a1 |
21.1-a |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{32} \cdot 7^{4} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1.046149643$ |
$17.48381656$ |
2.98186 |
\( \frac{95456569237420574}{2109289329} a^{4} - \frac{190913138474841148}{2109289329} a^{3} + \frac{31442431223779577}{2109289329} a^{2} + \frac{3048292286363857}{100442349} a - \frac{16488371703532606}{2109289329} \) |
\( \bigl[-2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 3\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 8 a^{2} - a + 3\) , \( 0\) , \( -59 a^{4} + 118 a^{3} + 4 a^{2} - 63 a + 11\) , \( -690 a^{4} + 1380 a^{3} + 68 a^{2} - 758 a + 161\bigr] \) |
${y}^2+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+3\right){x}{y}={x}^{3}+\left(-a^{5}+3a^{4}+2a^{3}-8a^{2}-a+3\right){x}^{2}+\left(-59a^{4}+118a^{3}+4a^{2}-63a+11\right){x}-690a^{4}+1380a^{3}+68a^{2}-758a+161$ |
| 21.1-a2 |
21.1-a |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.261537410$ |
$4475.857041$ |
2.98186 |
\( \frac{897469}{3969} a^{4} - \frac{1794938}{3969} a^{3} - \frac{4300136}{3969} a^{2} + \frac{247505}{189} a + \frac{6651403}{3969} \) |
\( \bigl[-2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 3\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 8 a^{2} - a + 3\) , \( 0\) , \( a^{4} - 2 a^{3} - a^{2} + 2 a + 1\) , \( 0\bigr] \) |
${y}^2+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+3\right){x}{y}={x}^{3}+\left(-a^{5}+3a^{4}+2a^{3}-8a^{2}-a+3\right){x}^{2}+\left(a^{4}-2a^{3}-a^{2}+2a+1\right){x}$ |
| 21.1-a3 |
21.1-a |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7^{8} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.523074821$ |
$1118.964260$ |
2.98186 |
\( \frac{2469125117705}{15752961} a^{4} - \frac{4938250235410}{15752961} a^{3} - \frac{10256295035167}{15752961} a^{2} + \frac{605972388232}{750141} a + \frac{12380135655164}{15752961} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{4} + 2 a^{3} + 2 a^{2} - 3 a - 1\) , \( a^{2} - a - 1\) , \( 13 a^{4} - 26 a^{3} - 35 a^{2} + 48 a - 14\) , \( 39 a^{4} - 78 a^{3} - 106 a^{2} + 145 a - 32\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+2a^{2}-3a-1\right){x}^{2}+\left(13a^{4}-26a^{3}-35a^{2}+48a-14\right){x}+39a^{4}-78a^{3}-106a^{2}+145a-32$ |
| 21.1-a4 |
21.1-a |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{32} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$2.092299286$ |
$69.93526626$ |
2.98186 |
\( -\frac{7508368686128970279577}{299096375126409} a^{4} + \frac{15016737372257940559154}{299096375126409} a^{3} + \frac{20908516177655160718196}{299096375126409} a^{2} - \frac{1353184993513530047513}{14242684529829} a + \frac{5658133340658671444093}{299096375126409} \) |
\( \bigl[-2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 3\) , \( -a^{5} + 3 a^{4} + 2 a^{3} - 8 a^{2} - a + 3\) , \( 0\) , \( -159 a^{4} + 318 a^{3} + 79 a^{2} - 238 a - 69\) , \( 1892 a^{4} - 3784 a^{3} + 296 a^{2} + 1596 a - 1030\bigr] \) |
${y}^2+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+3\right){x}{y}={x}^{3}+\left(-a^{5}+3a^{4}+2a^{3}-8a^{2}-a+3\right){x}^{2}+\left(-159a^{4}+318a^{3}+79a^{2}-238a-69\right){x}+1892a^{4}-3784a^{3}+296a^{2}+1596a-1030$ |
| 21.1-a5 |
21.1-a |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{16} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.046149643$ |
$279.7410650$ |
2.98186 |
\( \frac{69292788128748168482}{466948881} a^{4} - \frac{138585576257496336964}{466948881} a^{3} - \frac{290939024069228830105}{466948881} a^{2} + \frac{17153895818951285647}{22235661} a + \frac{348746065681006191566}{466948881} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{4} + 2 a^{3} + 2 a^{2} - 3 a - 1\) , \( a^{2} - a - 1\) , \( 53 a^{4} - 106 a^{3} - 130 a^{2} + 183 a - 99\) , \( -351 a^{4} + 702 a^{3} + 990 a^{2} - 1341 a + 125\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+2a^{2}-3a-1\right){x}^{2}+\left(53a^{4}-106a^{3}-130a^{2}+183a-99\right){x}-351a^{4}+702a^{3}+990a^{2}-1341a+125$ |
| 21.1-a6 |
21.1-a |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{4} \) |
$2.092299286$ |
$4.370954141$ |
2.98186 |
\( \frac{2905242808362156164576920441}{21609} a^{4} - \frac{5810485616724312329153840882}{21609} a^{3} - \frac{12198217539758009162689930484}{21609} a^{2} + \frac{719212397529531682250802425}{1029} a + \frac{14621896551411174957571933555}{21609} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{4} + 2 a^{3} + 2 a^{2} - 3 a - 1\) , \( a^{2} - a - 1\) , \( -232 a^{4} + 464 a^{3} + 865 a^{2} - 1097 a - 739\) , \( -3520 a^{4} + 7040 a^{3} + 12262 a^{2} - 15782 a - 7870\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+2a^{2}-3a-1\right){x}^{2}+\left(-232a^{4}+464a^{3}+865a^{2}-1097a-739\right){x}-3520a^{4}+7040a^{3}+12262a^{2}-15782a-7870$ |
| 21.1-b1 |
21.1-b |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{32} \cdot 7^{4} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.586597879$ |
$1340.080365$ |
4.00479 |
\( \frac{95456569237420574}{2109289329} a^{4} - \frac{190913138474841148}{2109289329} a^{3} + \frac{31442431223779577}{2109289329} a^{2} + \frac{3048292286363857}{100442349} a - \frac{16488371703532606}{2109289329} \) |
\( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 3 a + 1\) , \( a^{2} - a - 2\) , \( -2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 4\) , \( a^{5} + 8 a^{4} - 25 a^{3} - 36 a^{2} + 59 a - 18\) , \( -51 a^{4} + 102 a^{3} + 161 a^{2} - 213 a + 38\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-2a^{2}+3a+1\right){x}{y}+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+4\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{5}+8a^{4}-25a^{3}-36a^{2}+59a-18\right){x}-51a^{4}+102a^{3}+161a^{2}-213a+38$ |
| 21.1-b2 |
21.1-b |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.146649469$ |
$21441.28584$ |
4.00479 |
\( \frac{897469}{3969} a^{4} - \frac{1794938}{3969} a^{3} - \frac{4300136}{3969} a^{2} + \frac{247505}{189} a + \frac{6651403}{3969} \) |
\( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 3 a + 1\) , \( a^{2} - a - 2\) , \( -2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 4\) , \( a^{5} - 2 a^{4} - 5 a^{3} + 9 a^{2} + 4 a - 3\) , \( a^{4} - 2 a^{3} - 1\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-2a^{2}+3a+1\right){x}{y}+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+4\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{5}-2a^{4}-5a^{3}+9a^{2}+4a-3\right){x}+a^{4}-2a^{3}-1$ |
| 21.1-b3 |
21.1-b |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7^{8} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.293298939$ |
$5360.321460$ |
4.00479 |
\( \frac{2469125117705}{15752961} a^{4} - \frac{4938250235410}{15752961} a^{3} - \frac{10256295035167}{15752961} a^{2} + \frac{605972388232}{750141} a + \frac{12380135655164}{15752961} \) |
\( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 3 a + 1\) , \( a^{2} - a - 2\) , \( -2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 4\) , \( a^{5} - 2 a^{4} - 5 a^{3} + 9 a^{2} + 4 a - 8\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 6 a - 2\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-2a^{2}+3a+1\right){x}{y}+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+4\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{5}-2a^{4}-5a^{3}+9a^{2}+4a-8\right){x}-a^{4}+2a^{3}+4a^{2}-6a-2$ |
| 21.1-b4 |
21.1-b |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{32} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$256$ |
\( 2^{3} \) |
$1.173195758$ |
$1.308672231$ |
4.00479 |
\( -\frac{7508368686128970279577}{299096375126409} a^{4} + \frac{15016737372257940559154}{299096375126409} a^{3} + \frac{20908516177655160718196}{299096375126409} a^{2} - \frac{1353184993513530047513}{14242684529829} a + \frac{5658133340658671444093}{299096375126409} \) |
\( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 3 a + 1\) , \( a^{2} - a - 2\) , \( -2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 4\) , \( a^{5} + 48 a^{4} - 105 a^{3} - 151 a^{2} + 214 a - 133\) , \( 373 a^{4} - 746 a^{3} - 1290 a^{2} + 1662 a - 613\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-2a^{2}+3a+1\right){x}{y}+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+4\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{5}+48a^{4}-105a^{3}-151a^{2}+214a-133\right){x}+373a^{4}-746a^{3}-1290a^{2}+1662a-613$ |
| 21.1-b5 |
21.1-b |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{16} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2^{4} \) |
$0.586597879$ |
$83.75502282$ |
4.00479 |
\( \frac{69292788128748168482}{466948881} a^{4} - \frac{138585576257496336964}{466948881} a^{3} - \frac{290939024069228830105}{466948881} a^{2} + \frac{17153895818951285647}{22235661} a + \frac{348746065681006191566}{466948881} \) |
\( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 3 a + 1\) , \( a^{2} - a - 2\) , \( -2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 4\) , \( a^{5} - 12 a^{4} + 15 a^{3} + 54 a^{2} - 51 a - 78\) , \( -39 a^{4} + 78 a^{3} + 163 a^{2} - 203 a - 226\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-2a^{2}+3a+1\right){x}{y}+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+4\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{5}-12a^{4}+15a^{3}+54a^{2}-51a-78\right){x}-39a^{4}+78a^{3}+163a^{2}-203a-226$ |
| 21.1-b6 |
21.1-b |
$6$ |
$8$ |
6.6.1387029.1 |
$6$ |
$[6, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$135.63328$ |
$(a^5-3a^4-2a^3+8a^2+a-2), (a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
✓ |
✓ |
|
$2$ |
2B |
$256$ |
\( 2^{3} \) |
$1.173195758$ |
$1.308672231$ |
4.00479 |
\( \frac{2905242808362156164576920441}{21609} a^{4} - \frac{5810485616724312329153840882}{21609} a^{3} - \frac{12198217539758009162689930484}{21609} a^{2} + \frac{719212397529531682250802425}{1029} a + \frac{14621896551411174957571933555}{21609} \) |
\( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 3 a + 1\) , \( a^{2} - a - 2\) , \( -2 a^{5} + 5 a^{4} + 6 a^{3} - 13 a^{2} - 4 a + 4\) , \( a^{5} - 232 a^{4} + 455 a^{3} + 979 a^{2} - 1196 a - 1143\) , \( -3103 a^{4} + 6206 a^{3} + 13092 a^{2} - 16196 a - 15855\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-2a^{2}+3a+1\right){x}{y}+\left(-2a^{5}+5a^{4}+6a^{3}-13a^{2}-4a+4\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(a^{5}-232a^{4}+455a^{3}+979a^{2}-1196a-1143\right){x}-3103a^{4}+6206a^{3}+13092a^{2}-16196a-15855$ |
*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.
