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 |
475.2-a1 |
475.2-a |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{6} \cdot 19^{2} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.125109672$ |
$316.7797274$ |
4.428853540 |
\( \frac{97722368}{361} a^{3} - \frac{362807296}{361} a^{2} + \frac{32821248}{361} a + \frac{630816768}{361} \) |
\( \bigl[0\) , \( a^{3} - 6 a - 5\) , \( a^{2} - 3\) , \( -6 a^{3} - 2 a^{2} + 30 a + 27\) , \( -3 a^{3} - 4 a^{2} + 6 a + 7\bigr] \) |
${y}^2+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-6a-5\right){x}^{2}+\left(-6a^{3}-2a^{2}+30a+27\right){x}-3a^{3}-4a^{2}+6a+7$ |
475.2-b1 |
475.2-b |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( - 5^{7} \cdot 19 \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$256$ |
\( 2 \) |
$1$ |
$0.918607856$ |
1.642453905 |
\( \frac{1696880810079666557574464889868927}{95} a^{3} + \frac{1198612397219876598057410152658112}{95} a^{2} - \frac{6937405622051853730571323680910636}{95} a - \frac{6896966405593066660677049547422819}{95} \) |
\( \bigl[a^{2} - a - 4\) , \( a^{3} - 5 a - 5\) , \( 1\) , \( -72 a^{3} - 569 a^{2} + 477 a + 2106\) , \( 4469 a^{3} - 29555 a^{2} - 4776 a + 100127\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a-5\right){x}^{2}+\left(-72a^{3}-569a^{2}+477a+2106\right){x}+4469a^{3}-29555a^{2}-4776a+100127$ |
475.2-b2 |
475.2-b |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( - 5^{7} \cdot 19 \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$64$ |
\( 2 \) |
$1$ |
$3.674431424$ |
1.642453905 |
\( -\frac{22506195349366937817727}{95} a^{3} - \frac{2753261774705929528512}{95} a^{2} + \frac{129193832172995183144236}{95} a + \frac{116649031622658518623379}{95} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( 1\) , \( -320 a^{3} + 725 a^{2} + 1089 a - 2852\) , \( 5157 a^{3} - 19101 a^{2} - 15730 a + 67990\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a-1\right){x}^{2}+\left(-320a^{3}+725a^{2}+1089a-2852\right){x}+5157a^{3}-19101a^{2}-15730a+67990$ |
475.2-b3 |
475.2-b |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{14} \cdot 19^{8} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$14.69772569$ |
1.642453905 |
\( \frac{796679597897812537}{424589076025} a^{3} - \frac{496789503283374063}{84917815205} a^{2} - \frac{1998336689606502628}{424589076025} a + \frac{7814363453289350341}{424589076025} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( 1\) , \( -70 a^{3} + 195 a^{2} + 194 a - 622\) , \( -601 a^{3} + 1781 a^{2} + 1554 a - 5574\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a-1\right){x}^{2}+\left(-70a^{3}+195a^{2}+194a-622\right){x}-601a^{3}+1781a^{2}+1554a-5574$ |
475.2-b4 |
475.2-b |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{10} \cdot 19^{4} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$58.79090279$ |
1.642453905 |
\( \frac{1518174993681}{130321} a^{3} + \frac{5371675814913}{651605} a^{2} - \frac{31003619598681}{651605} a - \frac{6165675735772}{130321} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( 1\) , \( -10 a^{3} + 10 a^{2} + 34 a - 22\) , \( -33 a^{3} + 9 a^{2} + 120 a + 20\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a-1\right){x}^{2}+\left(-10a^{3}+10a^{2}+34a-22\right){x}-33a^{3}+9a^{2}+120a+20$ |
475.2-b5 |
475.2-b |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{8} \cdot 19^{2} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$58.79090279$ |
1.642453905 |
\( -\frac{1944419}{1805} a^{3} - \frac{149040}{361} a^{2} + \frac{6980213}{1805} a + \frac{7045469}{1805} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( 1\) , \( -a + 3\) , \( -a^{3} + a^{2} + 4 a - 4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a-1\right){x}^{2}+\left(-a+3\right){x}-a^{3}+a^{2}+4a-4$ |
475.2-b6 |
475.2-b |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{8} \cdot 19^{2} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$16$ |
\( 2^{3} \) |
$1$ |
$14.69772569$ |
1.642453905 |
\( \frac{2428529195211826419}{1805} a^{3} + \frac{343085431395379515}{361} a^{2} - \frac{9928607289091056588}{1805} a - \frac{9870737298139211129}{1805} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 3 a - 1\) , \( 1\) , \( -110 a^{3} - 15 a^{2} + 434 a + 178\) , \( -1153 a^{3} - 1131 a^{2} + 4770 a + 5970\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a-1\right){x}^{2}+\left(-110a^{3}-15a^{2}+434a+178\right){x}-1153a^{3}-1131a^{2}+4770a+5970$ |
475.2-c1 |
475.2-c |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( - 5^{7} \cdot 19 \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2 \) |
$0.793129353$ |
$12.38196187$ |
4.389717068 |
\( -\frac{22506195349366937817727}{95} a^{3} - \frac{2753261774705929528512}{95} a^{2} + \frac{129193832172995183144236}{95} a + \frac{116649031622658518623379}{95} \) |
\( \bigl[a^{3} - 4 a - 4\) , \( a\) , \( a^{3} - 5 a - 3\) , \( 2301 a^{3} - 3350 a^{2} - 7454 a - 984\) , \( -14655 a^{3} - 138204 a^{2} + 300315 a + 490111\bigr] \) |
${y}^2+\left(a^{3}-4a-4\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+a{x}^{2}+\left(2301a^{3}-3350a^{2}-7454a-984\right){x}-14655a^{3}-138204a^{2}+300315a+490111$ |
475.2-c2 |
475.2-c |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{8} \cdot 19^{2} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$0.396564676$ |
$99.05569501$ |
4.389717068 |
\( \frac{2428529195211826419}{1805} a^{3} + \frac{343085431395379515}{361} a^{2} - \frac{9928607289091056588}{1805} a - \frac{9870737298139211129}{1805} \) |
\( \bigl[a^{3} - 4 a - 4\) , \( a\) , \( a^{3} - 5 a - 3\) , \( 136 a^{3} - 215 a^{2} - 434 a - 29\) , \( -135 a^{3} - 2109 a^{2} + 4120 a + 7086\bigr] \) |
${y}^2+\left(a^{3}-4a-4\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+a{x}^{2}+\left(136a^{3}-215a^{2}-434a-29\right){x}-135a^{3}-2109a^{2}+4120a+7086$ |
475.2-c3 |
475.2-c |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{8} \cdot 19^{2} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.099141169$ |
$396.2227800$ |
4.389717068 |
\( -\frac{1944419}{1805} a^{3} - \frac{149040}{361} a^{2} + \frac{6980213}{1805} a + \frac{7045469}{1805} \) |
\( \bigl[a^{3} - 4 a - 4\) , \( a\) , \( a^{3} - 5 a - 3\) , \( a^{3} + 5 a^{2} - 9 a - 14\) , \( 3 a^{3} - a^{2} - 9 a - 5\bigr] \) |
${y}^2+\left(a^{3}-4a-4\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+a{x}^{2}+\left(a^{3}+5a^{2}-9a-14\right){x}+3a^{3}-a^{2}-9a-5$ |
475.2-c4 |
475.2-c |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{10} \cdot 19^{4} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.198282338$ |
$396.2227800$ |
4.389717068 |
\( \frac{1518174993681}{130321} a^{3} + \frac{5371675814913}{651605} a^{2} - \frac{31003619598681}{651605} a - \frac{6165675735772}{130321} \) |
\( \bigl[a^{3} - 4 a - 4\) , \( a\) , \( a^{3} - 5 a - 3\) , \( 11 a^{3} - 15 a^{2} - 34 a - 4\) , \( -14 a^{2} + 25 a + 46\bigr] \) |
${y}^2+\left(a^{3}-4a-4\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+a{x}^{2}+\left(11a^{3}-15a^{2}-34a-4\right){x}-14a^{2}+25a+46$ |
475.2-c5 |
475.2-c |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{14} \cdot 19^{8} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.099141169$ |
$99.05569501$ |
4.389717068 |
\( \frac{796679597897812537}{424589076025} a^{3} - \frac{496789503283374063}{84917815205} a^{2} - \frac{1998336689606502628}{424589076025} a + \frac{7814363453289350341}{424589076025} \) |
\( \bigl[a^{3} - 4 a - 4\) , \( a\) , \( a^{3} - 5 a - 3\) , \( 46 a^{3} - 135 a^{2} - 34 a + 181\) , \( -277 a^{3} + 1089 a^{2} - 194 a - 1950\bigr] \) |
${y}^2+\left(a^{3}-4a-4\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+a{x}^{2}+\left(46a^{3}-135a^{2}-34a+181\right){x}-277a^{3}+1089a^{2}-194a-1950$ |
475.2-c6 |
475.2-c |
$6$ |
$8$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( - 5^{7} \cdot 19 \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2 \) |
$0.793129353$ |
$12.38196187$ |
4.389717068 |
\( \frac{1696880810079666557574464889868927}{95} a^{3} + \frac{1198612397219876598057410152658112}{95} a^{2} - \frac{6937405622051853730571323680910636}{95} a - \frac{6896966405593066660677049547422819}{95} \) |
\( \bigl[a^{2} - 3\) , \( a^{3} - 2 a^{2} - 2 a + 4\) , \( a\) , \( 573 a^{3} + 17 a^{2} - 3271 a - 2848\) , \( -14446 a^{3} - 1709 a^{2} + 83121 a + 74667\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-2a^{2}-2a+4\right){x}^{2}+\left(573a^{3}+17a^{2}-3271a-2848\right){x}-14446a^{3}-1709a^{2}+83121a+74667$ |
475.2-d1 |
475.2-d |
$1$ |
$1$ |
4.4.5125.1 |
$4$ |
$[4, 0]$ |
475.2 |
\( 5^{2} \cdot 19 \) |
\( 5^{6} \cdot 19^{2} \) |
$13.82204$ |
$(-a^3+a^2+4a+1), (2a^2-a-6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$72.31627871$ |
2.020315197 |
\( \frac{97722368}{361} a^{3} - \frac{362807296}{361} a^{2} + \frac{32821248}{361} a + \frac{630816768}{361} \) |
\( \bigl[0\) , \( -a^{2} + 2 a + 5\) , \( a^{2} - a - 4\) , \( 3 a^{3} - 15 a^{2} + 12 a + 21\) , \( 9 a^{3} - 38 a^{2} + 21 a + 44\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{2}+2a+5\right){x}^{2}+\left(3a^{3}-15a^{2}+12a+21\right){x}+9a^{3}-38a^{2}+21a+44$ |
*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.