| 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 |
| 1.1-a1 |
1.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 3^{12} \) |
$1.35225$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3, 5$ |
2Cn, 3B.1.2, 5B |
$9$ |
\( 1 \) |
$1$ |
$4.286906993$ |
2.549581091 |
\( -299510191348095 a + 2415960913292737 \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 22 a - 285\) , \( 307 a - 3606\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(22a-285\right){x}+307a-3606$ |
| 1.1-a2 |
1.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 3^{12} \) |
$1.35225$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3, 5$ |
2Cn, 3B.1.1, 5B |
$9$ |
\( 1 \) |
$1$ |
$38.58216293$ |
2.549581091 |
\( -888615 a + 7334137 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( a - 3\) , \( 2 a + 6\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(a-3\right){x}+2a+6$ |
| 1.1-a3 |
1.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 3^{12} \) |
$1.35225$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3, 5$ |
2Cn, 3B.1.1, 5B |
$9$ |
\( 1 \) |
$1$ |
$38.58216293$ |
2.549581091 |
\( 888615 a + 6445522 \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -3 a\) , \( -3 a + 9\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-3a{x}-3a+9$ |
| 1.1-a4 |
1.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 3^{12} \) |
$1.35225$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2, 3, 5$ |
2Cn, 3B.1.2, 5B |
$9$ |
\( 1 \) |
$1$ |
$4.286906993$ |
2.549581091 |
\( 299510191348095 a + 2116450721944642 \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -24 a - 263\) , \( -308 a - 3299\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-24a-263\right){x}-308a-3299$ |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \cdot 3^{12} \) |
$1.91237$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$0.503855175$ |
$29.61538193$ |
3.944257969 |
\( \frac{39151}{4} a + \frac{143637}{2} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -820 a - 5692\) , \( -36720 a - 259213\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-820a-5692\right){x}-36720a-259213$ |
| 4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{16} \) |
$1.91237$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$1$ |
$24.10357419$ |
3.185618033 |
\( \frac{47163267397}{256} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -1137 a - 7919\) , \( 53324 a + 377114\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1137a-7919\right){x}+53324a+377114$ |
| 4.1-c1 |
4.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \cdot 3^{12} \) |
$1.91237$ |
$(2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2 \) |
$0.503855175$ |
$29.61538193$ |
3.944257969 |
\( -\frac{39151}{4} a + \frac{326425}{4} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -26 a - 78\) , \( a + 273\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-26a-78\right){x}+a+273$ |
| 4.1-d1 |
4.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{4} \) |
$1.91237$ |
$(2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cn, 5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$61.34905438$ |
0.324324770 |
\( \frac{226981}{4} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -26 a - 94\) , \( 62 a + 662\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-26a-94\right){x}+62a+662$ |
| 4.1-d2 |
4.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{20} \) |
$1.91237$ |
$(2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cn, 5B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.453962175$ |
0.324324770 |
\( \frac{71009375221}{1024} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -1301 a - 9104\) , \( -72823 a - 514373\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1301a-9104\right){x}-72823a-514373$ |
| 5.1-a1 |
5.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 3^{12} \cdot 5^{9} \) |
$2.02208$ |
$(5,a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$1.729302047$ |
1.028479463 |
\( -\frac{172505409481}{1953125} a - \frac{1232537907253}{1953125} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 2012 a - 16219\) , \( 162388 a - 1309832\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(2012a-16219\right){x}+162388a-1309832$ |
| 5.1-a2 |
5.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
5.1 |
\( 5 \) |
\( 3^{12} \cdot 5^{3} \) |
$2.02208$ |
$(5,a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$9$ |
\( 1 \) |
$1$ |
$15.56371843$ |
1.028479463 |
\( -\frac{12979}{125} a + \frac{115898}{125} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -168 a + 1366\) , \( -1529 a + 12385\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-168a+1366\right){x}-1529a+12385$ |
| 5.2-a1 |
5.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 3^{12} \cdot 5^{9} \) |
$2.02208$ |
$(5,a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$1.729302047$ |
1.028479463 |
\( \frac{172505409481}{1953125} a - \frac{1405043316734}{1953125} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -2014 a - 14205\) , \( -162389 a - 1147443\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-2014a-14205\right){x}-162389a-1147443$ |
| 5.2-a2 |
5.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
5.2 |
\( 5 \) |
\( 3^{12} \cdot 5^{3} \) |
$2.02208$ |
$(5,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$9$ |
\( 1 \) |
$1$ |
$15.56371843$ |
1.028479463 |
\( \frac{12979}{125} a + \frac{102919}{125} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 166 a + 1200\) , \( 1528 a + 10857\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(166a+1200\right){x}+1528a+10857$ |
| 9.2-a1 |
9.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{18} \) |
$2.34216$ |
$(3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3, 5$ |
2Cn, 3B, 5B |
$1$ |
\( 1 \) |
$8.585789875$ |
$2.840136404$ |
3.222787782 |
\( -299510191348095 a + 2415960913292737 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 755121 a - 6091088\) , \( 982309525 a - 7923675006\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(755121a-6091088\right){x}+982309525a-7923675006$ |
| 9.2-a2 |
9.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.34216$ |
$(3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3, 5$ |
2Cn, 3B, 5B |
$1$ |
\( 1 \) |
$0.572385991$ |
$42.60204606$ |
3.222787782 |
\( -888615 a + 7334137 \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 23 a - 21\) , \( -a + 523\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(23a-21\right){x}-a+523$ |
| 9.2-a3 |
9.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{18} \) |
$2.34216$ |
$(3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3, 5$ |
2Cn, 3B, 5B |
$1$ |
\( 1 \) |
$2.861929958$ |
$8.520409212$ |
3.222787782 |
\( 888615 a + 6445522 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 9336 a - 75308\) , \( 1341295 a - 10819386\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(9336a-75308\right){x}+1341295a-10819386$ |
| 9.2-a4 |
9.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.34216$ |
$(3,a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3, 5$ |
2Cn, 3B, 5B |
$1$ |
\( 1 \) |
$1.717157975$ |
$14.20068202$ |
3.222787782 |
\( 299510191348095 a + 2116450721944642 \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 33 a - 126\) , \( 144 a - 797\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(33a-126\right){x}+144a-797$ |
| 9.3-a1 |
9.3-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.34216$ |
$(3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3, 5$ |
2Cn, 3B, 5B |
$1$ |
\( 1 \) |
$1.717157975$ |
$14.20068202$ |
3.222787782 |
\( -299510191348095 a + 2415960913292737 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -3 a - 64\) , \( -241 a - 1693\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-3a-64\right){x}-241a-1693$ |
| 9.3-a2 |
9.3-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{18} \) |
$2.34216$ |
$(3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3, 5$ |
2Cn, 3B, 5B |
$1$ |
\( 1 \) |
$2.861929958$ |
$8.520409212$ |
3.222787782 |
\( -888615 a + 7334137 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -9336 a - 65972\) , \( -1341295 a - 9478091\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-9336a-65972\right){x}-1341295a-9478091$ |
| 9.3-a3 |
9.3-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.34216$ |
$(3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3, 5$ |
2Cn, 3B, 5B |
$1$ |
\( 1 \) |
$0.572385991$ |
$42.60204606$ |
3.222787782 |
\( 888615 a + 6445522 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 7 a + 31\) , \( 9 a + 52\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7a+31\right){x}+9a+52$ |
| 9.3-a4 |
9.3-a |
$4$ |
$15$ |
\(\Q(\sqrt{229}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{18} \) |
$2.34216$ |
$(3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2, 3, 5$ |
2Cn, 3B, 5B |
$1$ |
\( 1 \) |
$8.585789875$ |
$2.840136404$ |
3.222787782 |
\( 299510191348095 a + 2116450721944642 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -755121 a - 5335967\) , \( -982309525 a - 6941365481\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-755121a-5335967\right){x}-982309525a-6941365481$ |
*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.
