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
8.1-a1
8.1-a
$4$
$21$
\(\Q(\zeta_{9})^+\)
$3$
$[3, 0]$
8.1
\( 2^{3} \)
\( - 2^{9} \)
$1.13736$
$(2)$
0
$\Z/21\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$3, 7$
3B.1.1 , 7B.1.1
$1$
\( 3 \)
$1$
$386.8008344$
0.292366465
\( -\frac{140625}{8} \)
\( \bigl[a^{2} + a - 1\) , \( -a^{2} + a + 2\) , \( a^{2} - 2\) , \( 13 a^{2} + 2 a - 44\) , \( -22 a^{2} - 3 a + 88\bigr] \)
${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(13a^{2}+2a-44\right){x}-22a^{2}-3a+88$
64.1-a2
64.1-a
$4$
$21$
6.6.820125.1
$6$
$[6, 0]$
64.1
\( 2^{6} \)
\( 2^{18} \)
$114.44414$
$(2)$
$1$
$\Z/21\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$3, 7$
3B.1.1 , 7B.1.1
$1$
\( 3 \)
$0.541610385$
$149614.8855$
3.65221
\( -\frac{140625}{8} \)
\( \bigl[\frac{8}{19} a^{5} + \frac{1}{19} a^{4} - \frac{60}{19} a^{3} - \frac{49}{19} a^{2} - \frac{3}{19} a + \frac{26}{19}\) , \( \frac{4}{19} a^{5} - \frac{9}{19} a^{4} - \frac{30}{19} a^{3} + \frac{61}{19} a^{2} + \frac{27}{19} a - \frac{25}{19}\) , \( \frac{6}{19} a^{5} - \frac{4}{19} a^{4} - \frac{45}{19} a^{3} + \frac{6}{19} a^{2} + \frac{12}{19} a - \frac{9}{19}\) , \( \frac{8}{19} a^{5} + \frac{1}{19} a^{4} - \frac{60}{19} a^{3} - \frac{49}{19} a^{2} - \frac{3}{19} a + \frac{7}{19}\) , \( 0\bigr] \)
${y}^2+\left(\frac{8}{19}a^{5}+\frac{1}{19}a^{4}-\frac{60}{19}a^{3}-\frac{49}{19}a^{2}-\frac{3}{19}a+\frac{26}{19}\right){x}{y}+\left(\frac{6}{19}a^{5}-\frac{4}{19}a^{4}-\frac{45}{19}a^{3}+\frac{6}{19}a^{2}+\frac{12}{19}a-\frac{9}{19}\right){y}={x}^{3}+\left(\frac{4}{19}a^{5}-\frac{9}{19}a^{4}-\frac{30}{19}a^{3}+\frac{61}{19}a^{2}+\frac{27}{19}a-\frac{25}{19}\right){x}^{2}+\left(\frac{8}{19}a^{5}+\frac{1}{19}a^{4}-\frac{60}{19}a^{3}-\frac{49}{19}a^{2}-\frac{3}{19}a+\frac{7}{19}\right){x}$
8.1-b2
8.1-b
$4$
$21$
\(\Q(\zeta_{36})^+\)
$6$
$[6, 0]$
8.1
\( 2^{3} \)
\( 2^{18} \)
$119.27013$
$(-a^4-a^3+3a^2+2a-1)$
$1$
$\Z/21\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$3, 7$
3B.1.1 , 7B.1.1
$1$
\( 2 \cdot 3 \)
$0.146674927$
$149614.8855$
1.59609
\( -\frac{140625}{8} \)
\( \bigl[a^{4} - 5 a^{2} + 4\) , \( -a^{4} + 6 a^{2} - 7\) , \( a^{2} - 2\) , \( a^{4} - 8 a^{2} + 9\) , \( 2 a^{2} - 3\bigr] \)
${y}^2+\left(a^{4}-5a^{2}+4\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{4}+6a^{2}-7\right){x}^{2}+\left(a^{4}-8a^{2}+9\right){x}+2a^{2}-3$
64.1-b2
64.1-b
$4$
$21$
6.6.1292517.1
$6$
$[6, 0]$
64.1
\( 2^{6} \)
\( 2^{18} \)
$143.67185$
$(2)$
$1$
$\Z/21\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$3, 7$
3B.1.1 , 7B.1.1
$1$
\( 3 \)
$0.755652443$
$149614.8855$
4.05894
\( -\frac{140625}{8} \)
\( \bigl[a^{5} - 6 a^{3} - a^{2} + 5 a\) , \( a^{5} - a^{4} - 6 a^{3} + 4 a^{2} + 6 a - 3\) , \( a^{4} - 5 a^{2} - a + 2\) , \( -a^{5} + 3 a^{4} + 6 a^{3} - 14 a^{2} - 8 a + 5\) , \( -2 a^{4} + 10 a^{2} + 2 a - 3\bigr] \)
${y}^2+\left(a^{5}-6a^{3}-a^{2}+5a\right){x}{y}+\left(a^{4}-5a^{2}-a+2\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+4a^{2}+6a-3\right){x}^{2}+\left(-a^{5}+3a^{4}+6a^{3}-14a^{2}-8a+5\right){x}-2a^{4}+10a^{2}+2a-3$
64.1-d2
64.1-d
$4$
$21$
6.6.1397493.1
$6$
$[6, 0]$
64.1
\( 2^{6} \)
\( 2^{18} \)
$149.39235$
$(2)$
$1$
$\Z/21\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$3, 7$
3B.1.1 , 7B.1.1
$1$
\( 3 \)
$0.408506482$
$149614.8855$
2.11024
\( -\frac{140625}{8} \)
\( \bigl[a^{5} - 3 a^{4} - 2 a^{3} + 7 a^{2} + 2 a - 1\) , \( a^{5} - 3 a^{4} - 4 a^{3} + 11 a^{2} + 6 a - 6\) , \( a^{3} - 2 a^{2} - 2 a + 3\) , \( -1\) , \( -4 a^{5} + 12 a^{4} + 14 a^{3} - 40 a^{2} - 20 a + 21\bigr] \)
${y}^2+\left(a^{5}-3a^{4}-2a^{3}+7a^{2}+2a-1\right){x}{y}+\left(a^{3}-2a^{2}-2a+3\right){y}={x}^{3}+\left(a^{5}-3a^{4}-4a^{3}+11a^{2}+6a-6\right){x}^{2}-{x}-4a^{5}+12a^{4}+14a^{3}-40a^{2}-20a+21$
64.1-g2
64.1-g
$4$
$21$
6.6.1528713.1
$6$
$[6, 0]$
64.1
\( 2^{6} \)
\( 2^{18} \)
$156.24874$
$(2a^5-6a^4-5a^3+12a^2+a-3), (-a^5+3a^4+3a^3-6a^2-2a+1)$
0
$\Z/21\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$3, 7$
3B.1.1 , 7B.1.1
$1$
\( 3^{2} \)
$1$
$149614.8855$
2.46954
\( -\frac{140625}{8} \)
\( \bigl[a^{5} - 4 a^{4} + 9 a^{2} - 2 a - 3\) , \( -a^{5} + 5 a^{4} - 3 a^{3} - 11 a^{2} + 6 a + 3\) , \( a^{4} - 3 a^{3} - 2 a^{2} + 4 a + 1\) , \( a^{5} - 7 a^{4} + 9 a^{3} + 15 a^{2} - 14 a - 7\) , \( 2 a^{4} - 6 a^{3} - 4 a^{2} + 8 a + 3\bigr] \)
${y}^2+\left(a^{5}-4a^{4}+9a^{2}-2a-3\right){x}{y}+\left(a^{4}-3a^{3}-2a^{2}+4a+1\right){y}={x}^{3}+\left(-a^{5}+5a^{4}-3a^{3}-11a^{2}+6a+3\right){x}^{2}+\left(a^{5}-7a^{4}+9a^{3}+15a^{2}-14a-7\right){x}+2a^{4}-6a^{3}-4a^{2}+8a+3$
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*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.