| 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 |
| 3.1-a1 |
3.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$1.92525$ |
$(a-8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$17.06231653$ |
2.084493730 |
\( \frac{8413184}{729} a - \frac{69804032}{729} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 86254973 a - 706027382\) , \( -1253232701878 a + 10258151770118\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(86254973a-706027382\right){x}-1253232701878a+10258151770118$ |
| 3.1-a2 |
3.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{2} \) |
$1.92525$ |
$(a-8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$17.06231653$ |
2.084493730 |
\( \frac{3828325378727936}{9} a - \frac{31336193750392832}{9} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 6995444283 a - 57260179252\) , \( -911183561967440 a + 7458358894634797\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(6995444283a-57260179252\right){x}-911183561967440a+7458358894634797$ |
| 3.1-b1 |
3.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{6} \) |
$1.92525$ |
$(a-8)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$2.040155743$ |
$10.61988195$ |
1.764632815 |
\( \frac{8413184}{729} a - \frac{69804032}{729} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( 86254973 a - 706027382\) , \( 1253232701878 a - 10258151770135\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(86254973a-706027382\right){x}+1253232701878a-10258151770135$ |
| 3.1-b2 |
3.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{2} \) |
$1.92525$ |
$(a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$6.120467229$ |
$1.179986884$ |
1.764632815 |
\( \frac{3828325378727936}{9} a - \frac{31336193750392832}{9} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( 6995444283 a - 57260179252\) , \( 911183561967440 a - 7458358894634814\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(6995444283a-57260179252\right){x}+911183561967440a-7458358894634814$ |
| 3.2-a1 |
3.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{2} \) |
$1.92525$ |
$(a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$17.06231653$ |
2.084493730 |
\( -\frac{3828325378727936}{9} a - \frac{31336193750392832}{9} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -6995444283 a - 57260179252\) , \( 911183561967440 a + 7458358894634797\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-6995444283a-57260179252\right){x}+911183561967440a+7458358894634797$ |
| 3.2-a2 |
3.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{6} \) |
$1.92525$ |
$(a+8)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$17.06231653$ |
2.084493730 |
\( -\frac{8413184}{729} a - \frac{69804032}{729} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -86254973 a - 706027382\) , \( 1253232701878 a + 10258151770118\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-86254973a-706027382\right){x}+1253232701878a+10258151770118$ |
| 3.2-b1 |
3.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{2} \) |
$1.92525$ |
$(a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$6.120467229$ |
$1.179986884$ |
1.764632815 |
\( -\frac{3828325378727936}{9} a - \frac{31336193750392832}{9} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -6995444283 a - 57260179252\) , \( -911183561967440 a - 7458358894634814\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(-6995444283a-57260179252\right){x}-911183561967440a-7458358894634814$ |
| 3.2-b2 |
3.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{6} \) |
$1.92525$ |
$(a+8)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$2.040155743$ |
$10.61988195$ |
1.764632815 |
\( -\frac{8413184}{729} a - \frac{69804032}{729} \) |
\( \bigl[0\) , \( 1\) , \( a\) , \( -86254973 a - 706027382\) , \( -1253232701878 a - 10258151770135\bigr] \) |
${y}^2+a{y}={x}^{3}+{x}^{2}+\left(-86254973a-706027382\right){x}-1253232701878a-10258151770135$ |
| 6.1-a1 |
6.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{29} \cdot 3^{4} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$3.141365875$ |
$2.159538622$ |
3.315141632 |
\( \frac{23975380097}{2654208} a + \frac{195281635615}{2654208} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 152614 a - 1249110\) , \( 223520503 a - 1829593995\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(152614a-1249110\right){x}+223520503a-1829593995$ |
| 6.1-b1 |
6.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.791190484$ |
$19.34653958$ |
3.740046019 |
\( \frac{7073}{18} a + \frac{57895}{18} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 60\) , \( a + 149\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+60{x}+a+149$ |
| 6.1-c1 |
6.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{6} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.161305102$ |
$10.10421332$ |
3.584146192 |
\( \frac{2619377}{23328} a + \frac{21486103}{23328} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -8 a + 137\) , \( 1000 a - 7995\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-8a+137\right){x}+1000a-7995$ |
| 6.1-c2 |
6.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.053768367$ |
$10.10421332$ |
3.584146192 |
\( \frac{412563047707}{147456} a + \frac{3376973797787}{147456} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 72 a - 518\) , \( -27880 a + 228397\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(72a-518\right){x}-27880a+228397$ |
| 6.1-d1 |
6.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{11} \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 11 \) |
$0.111418110$ |
$17.89376521$ |
5.358494650 |
\( \frac{3514753}{576} a + \frac{29083529}{576} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -a + 19\) , \( 2 a - 12\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a+19\right){x}+2a-12$ |
| 6.1-e1 |
6.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{11} \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.778457326$ |
$12.82361221$ |
2.439145914 |
\( \frac{3514753}{576} a + \frac{29083529}{576} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -11 a + 68\) , \( -32 a + 234\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-11a+68\right){x}-32a+234$ |
| 6.1-f1 |
6.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{6} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$2.599504528$ |
$17.96120008$ |
3.802745567 |
\( \frac{2619377}{23328} a + \frac{21486103}{23328} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -8 a + 65\) , \( -1048 a + 8578\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-8a+65\right){x}-1048a+8578$ |
| 6.1-f2 |
6.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$7.798513584$ |
$1.995688898$ |
3.802745567 |
\( \frac{412563047707}{147456} a + \frac{3376973797787}{147456} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 72 a - 590\) , \( 28312 a - 231744\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(72a-590\right){x}+28312a-231744$ |
| 6.1-g1 |
6.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.309566540$ |
$46.10892363$ |
3.487639538 |
\( \frac{7073}{18} a + \frac{57895}{18} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( -a + 8\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-a+8$ |
| 6.1-h1 |
6.1-h |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{29} \cdot 3^{4} \) |
$2.28952$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 29 \) |
$0.064952494$ |
$8.878661807$ |
4.086334747 |
\( \frac{23975380097}{2654208} a + \frac{195281635615}{2654208} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 152614 a - 1249203\) , \( -222604816 a + 1822098931\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(152614a-1249203\right){x}-222604816a+1822098931$ |
| 6.2-a1 |
6.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{29} \cdot 3^{4} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$3.141365875$ |
$2.159538622$ |
3.315141632 |
\( -\frac{23975380097}{2654208} a + \frac{195281635615}{2654208} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( -152615 a - 1249110\) , \( -223520503 a - 1829593995\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-152615a-1249110\right){x}-223520503a-1829593995$ |
| 6.2-b1 |
6.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.791190484$ |
$19.34653958$ |
3.740046019 |
\( -\frac{7073}{18} a + \frac{57895}{18} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 60\) , \( -a + 149\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+60{x}-a+149$ |
| 6.2-c1 |
6.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.053768367$ |
$10.10421332$ |
3.584146192 |
\( -\frac{412563047707}{147456} a + \frac{3376973797787}{147456} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -72 a - 518\) , \( 27880 a + 228397\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-72a-518\right){x}+27880a+228397$ |
| 6.2-c2 |
6.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{6} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.161305102$ |
$10.10421332$ |
3.584146192 |
\( -\frac{2619377}{23328} a + \frac{21486103}{23328} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 8 a + 137\) , \( -1000 a - 7995\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(8a+137\right){x}-1000a-7995$ |
| 6.2-d1 |
6.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{11} \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 11 \) |
$0.111418110$ |
$17.89376521$ |
5.358494650 |
\( -\frac{3514753}{576} a + \frac{29083529}{576} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( a + 19\) , \( -2 a - 12\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+19\right){x}-2a-12$ |
| 6.2-e1 |
6.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{11} \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.778457326$ |
$12.82361221$ |
2.439145914 |
\( -\frac{3514753}{576} a + \frac{29083529}{576} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 11 a + 68\) , \( 32 a + 234\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(11a+68\right){x}+32a+234$ |
| 6.2-f1 |
6.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{27} \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$7.798513584$ |
$1.995688898$ |
3.802745567 |
\( -\frac{412563047707}{147456} a + \frac{3376973797787}{147456} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -72 a - 590\) , \( -28312 a - 231744\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-72a-590\right){x}-28312a-231744$ |
| 6.2-f2 |
6.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{6} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$2.599504528$ |
$17.96120008$ |
3.802745567 |
\( -\frac{2619377}{23328} a + \frac{21486103}{23328} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 8 a + 65\) , \( 1048 a + 8578\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(8a+65\right){x}+1048a+8578$ |
| 6.2-g1 |
6.2-g |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.309566540$ |
$46.10892363$ |
3.487639538 |
\( -\frac{7073}{18} a + \frac{57895}{18} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( a + 8\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+a+8$ |
| 6.2-h1 |
6.2-h |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{29} \cdot 3^{4} \) |
$2.28952$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 29 \) |
$0.064952494$ |
$8.878661807$ |
4.086334747 |
\( -\frac{23975380097}{2654208} a + \frac{195281635615}{2654208} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( -152615 a - 1249203\) , \( 222604816 a + 1822098931\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-152615a-1249203\right){x}+222604816a+1822098931$ |
| 9.2-a1 |
9.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{3} \) |
$2.53377$ |
$(a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
0.902571723 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 24162 a + 197756\) , \( 243745 a + 1995129\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(24162a+197756\right){x}+243745a+1995129$ |
| 9.2-a2 |
9.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{3} \) |
$2.53377$ |
$(a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
0.902571723 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -259 a + 2421\) , \( -465 a + 4834\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-259a+2421\right){x}-465a+4834$ |
| 9.2-b1 |
9.2-b |
$2$ |
$67$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.53377$ |
$(a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-67$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$37.55116304$ |
$0.313676618$ |
2.878048675 |
\( -147197952000 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( -1760 a - 14410\) , \( -115010 a - 941417\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-1760a-14410\right){x}-115010a-941417$ |
| 9.2-b2 |
9.2-b |
$2$ |
$67$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.53377$ |
$(a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-67$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.560465120$ |
$21.01633344$ |
2.878048675 |
\( -147197952000 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -1760 a - 14410\) , \( 115010 a + 941400\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-1760a-14410\right){x}+115010a+941400$ |
| 9.3-a1 |
9.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{3} \) |
$2.53377$ |
$(a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
0.902571723 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 294 a + 2388\) , \( 2853 a + 23343\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(294a+2388\right){x}+2853a+23343$ |
| 9.3-a2 |
9.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{3} \) |
$2.53377$ |
$(a-8)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$29.55147182$ |
0.902571723 |
\( 1728 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -24127 a + 197789\) , \( -45989 a + 377464\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-24127a+197789\right){x}-45989a+377464$ |
| 9.3-b1 |
9.3-b |
$2$ |
$67$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.53377$ |
$(a-8)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-67$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
|
\( 2 \) |
$1$ |
$0.313676618$ |
2.878048675 |
\( -147197952000 \) |
\( \bigl[0\) , \( 0\) , \( a\) , \( 1760 a - 14410\) , \( 115010 a - 941417\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(1760a-14410\right){x}+115010a-941417$ |
| 9.3-b2 |
9.3-b |
$2$ |
$67$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{6} \) |
$2.53377$ |
$(a-8)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-67$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
|
\( 2 \) |
$1$ |
$21.01633344$ |
2.878048675 |
\( -147197952000 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 1760 a - 14410\) , \( -115010 a + 941400\bigr] \) |
${y}^2+{y}={x}^{3}+\left(1760a-14410\right){x}-115010a+941400$ |
| 12.1-a1 |
12.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$2.72271$ |
$(-27a+221), (a-8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$4.836114085$ |
$4.713393495$ |
5.569584918 |
\( \frac{1679360}{9} a - \frac{13746176}{9} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 3978 a - 32561\) , \( 389516 a - 3188347\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(3978a-32561\right){x}+389516a-3188347$ |
| 12.1-b1 |
12.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$2.72271$ |
$(-27a+221), (a-8)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.079655309$ |
$40.52794717$ |
4.732745839 |
\( \frac{1679360}{9} a - \frac{13746176}{9} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( 3978 a - 32561\) , \( -389517 a + 3188313\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(3978a-32561\right){x}-389517a+3188313$ |
| 12.2-a1 |
12.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$2.72271$ |
$(-27a+221), (a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$4.836114085$ |
$4.713393495$ |
5.569584918 |
\( -\frac{1679360}{9} a - \frac{13746176}{9} \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -3978 a - 32561\) , \( -389517 a - 3188347\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-3978a-32561\right){x}-389517a-3188347$ |
| 12.2-b1 |
12.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{2} \) |
$2.72271$ |
$(-27a+221), (a+8)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.079655309$ |
$40.52794717$ |
4.732745839 |
\( -\frac{1679360}{9} a - \frac{13746176}{9} \) |
\( \bigl[0\) , \( 1\) , \( a + 1\) , \( -3978 a - 32561\) , \( 389516 a + 3188313\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-3978a-32561\right){x}+389516a+3188313$ |
| 14.1-a1 |
14.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{8} \cdot 7 \) |
$2.82968$ |
$(-27a+221), (-11a-90)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.983708199$ |
$16.09214719$ |
3.867879021 |
\( -\frac{39956869}{112} a + \frac{161566453}{56} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -1258 a - 10268\) , \( 74011 a + 605837\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1258a-10268\right){x}+74011a+605837$ |
| 14.1-b1 |
14.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$2.82968$ |
$(-27a+221), (-11a-90)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$27.79705735$ |
3.395951051 |
\( -\frac{166357}{686} a + \frac{330363}{2744} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 262785 a - 2150932\) , \( -87350941 a + 714998420\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(262785a-2150932\right){x}-87350941a+714998420$ |
| 14.1-b2 |
14.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{9} \) |
$2.82968$ |
$(-27a+221), (-11a-90)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.088561928$ |
3.395951051 |
\( -\frac{46258413201758}{40353607} a + \frac{757367019175059}{80707214} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( 11005705 a - 90085522\) , \( 56919064821 a - 465902624852\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(11005705a-90085522\right){x}+56919064821a-465902624852$ |
| 14.1-c1 |
14.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$2.82968$ |
$(-27a+221), (-11a-90)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$9.899544131$ |
1.209421805 |
\( -\frac{166357}{686} a + \frac{330363}{2744} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 262785 a - 2150992\) , \( 88927653 a - 727904232\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(262785a-2150992\right){x}+88927653a-727904232$ |
| 14.1-c2 |
14.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{9} \) |
$2.82968$ |
$(-27a+221), (-11a-90)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$9.899544131$ |
1.209421805 |
\( -\frac{46258413201758}{40353607} a + \frac{757367019175059}{80707214} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 11005705 a - 90085582\) , \( -56853030589 a + 465362111500\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(11005705a-90085582\right){x}-56853030589a+465362111500$ |
| 14.1-d1 |
14.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{8} \cdot 7 \) |
$2.82968$ |
$(-27a+221), (-11a-90)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.365547038$ |
$11.95027030$ |
4.269466249 |
\( -\frac{39956869}{112} a + \frac{161566453}{56} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -1246 a - 10208\) , \( -81523 a - 667288\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-1246a-10208\right){x}-81523a-667288$ |
| 14.2-a1 |
14.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{8} \cdot 7 \) |
$2.82968$ |
$(-27a+221), (-11a+90)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.983708199$ |
$16.09214719$ |
3.867879021 |
\( \frac{39956869}{112} a + \frac{161566453}{56} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 1258 a - 10268\) , \( -74011 a + 605837\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(1258a-10268\right){x}-74011a+605837$ |
| 14.2-b1 |
14.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$2.82968$ |
$(-27a+221), (-11a+90)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$27.79705735$ |
3.395951051 |
\( \frac{166357}{686} a + \frac{330363}{2744} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -262786 a - 2150932\) , \( 87350940 a + 714998420\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-262786a-2150932\right){x}+87350940a+714998420$ |
| 14.2-b2 |
14.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{9} \) |
$2.82968$ |
$(-27a+221), (-11a+90)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.088561928$ |
3.395951051 |
\( \frac{46258413201758}{40353607} a + \frac{757367019175059}{80707214} \) |
\( \bigl[a\) , \( 0\) , \( a + 1\) , \( -11005706 a - 90085522\) , \( -56919064822 a - 465902624852\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-11005706a-90085522\right){x}-56919064822a-465902624852$ |
| 14.2-c1 |
14.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{67}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{3} \) |
$2.82968$ |
$(-27a+221), (-11a+90)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$9.899544131$ |
1.209421805 |
\( \frac{166357}{686} a + \frac{330363}{2744} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -262786 a - 2150992\) , \( -88927654 a - 727904232\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-262786a-2150992\right){x}-88927654a-727904232$ |
*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.
