| 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 |
| 2.2-a2 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( 2^{16} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$171.4772930$ |
0.602896961 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[a^{2} + a - 3\) , \( -a\) , \( a\) , \( -608022265892705467026233 a^{2} + 321836266424514724186808 a + 2583572058090078441238362\) , \( -627730820434767876188418741768082419 a^{2} + 332268331114001902947589285516929027 a + 2667316476142693080354081387519995775\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-608022265892705467026233a^{2}+321836266424514724186808a+2583572058090078441238362\right){x}-627730820434767876188418741768082419a^{2}+332268331114001902947589285516929027a+2667316476142693080354081387519995775$ |
| 16.5-b3 |
16.5-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
16.5 |
\( 2^{4} \) |
\( 2^{28} \) |
$2.52156$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$12.53690093$ |
0.705255777 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 2\) , \( -109074459285818 a^{2} + 57734919768569 a + 463472706625929\) , \( 1508268935406496095109 a^{2} - 798351754803931876384 a - 6408846675837241310282\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-109074459285818a^{2}+57734919768569a+463472706625929\right){x}+1508268935406496095109a^{2}-798351754803931876384a-6408846675837241310282$ |
| 32.1-c2 |
32.1-c |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{28} \) |
$2.83035$ |
$(a), (-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.264164657$ |
$30.31256343$ |
1.351372542 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[a\) , \( a^{2} - 3\) , \( a^{2} - 2\) , \( -77888706625301391598868 a^{2} + 41227783821563007723392 a + 330960060784047749989794\) , \( 28781005690695861341828735825476034 a^{2} - 15234263504867798437609722439797286 a - 122294538008473034126916564710880454\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-77888706625301391598868a^{2}+41227783821563007723392a+330960060784047749989794\right){x}+28781005690695861341828735825476034a^{2}-15234263504867798437609722439797286a-122294538008473034126916564710880454$ |
| 64.7-a5 |
64.7-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{34} \) |
$3.17696$ |
$(-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$28.64153339$ |
1.611212134 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{2} - 3\) , \( -2894790049560 a^{2} + 1532260369223 a + 12300369748958\) , \( -6521086825648874308 a^{2} + 3451719377275691295 a + 27709014383527172465\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-2894790049560a^{2}+1532260369223a+12300369748958\right){x}-6521086825648874308a^{2}+3451719377275691295a+27709014383527172465$ |
| 64.7-d3 |
64.7-d |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{34} \) |
$3.17696$ |
$(-a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.810264127$ |
$18.76465381$ |
2.565930457 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[a^{2} - 3\) , \( a^{2} - a - 4\) , \( a + 1\) , \( -2894790049560 a^{2} + 1532260369223 a + 12300369748958\) , \( 6521086825648874308 a^{2} - 3451719377275691296 a - 27709014383527172467\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-2894790049560a^{2}+1532260369223a+12300369748958\right){x}+6521086825648874308a^{2}-3451719377275691296a-27709014383527172467$ |
| 242.3-b7 |
242.3-b |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
242.3 |
\( 2 \cdot 11^{2} \) |
\( 2^{16} \cdot 11^{6} \) |
$3.96538$ |
$(-a+1), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$21.38345149$ |
1.202913127 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[1\) , \( a + 1\) , \( a^{2} - 2\) , \( -26140776857754976733142982 a^{2} + 13836746608257972575168441 a + 111075834644485451677761735\) , \( -176958717384573892178034799698113749293 a^{2} + 93667183109988451030481895839197063871 a + 751922459614203146023948231356362753108\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-26140776857754976733142982a^{2}+13836746608257972575168441a+111075834644485451677761735\right){x}-176958717384573892178034799698113749293a^{2}+93667183109988451030481895839197063871a+751922459614203146023948231356362753108$ |
*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.
