| 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 |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( - 3^{6} \) |
$2.37838$ |
$(11066a+106365)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.914998428$ |
$24.61936384$ |
3.341617801 |
\( \frac{60146}{729} a + \frac{885595}{729} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -422346227783811 a + 4481885041498204\) , \( -26439032262117056079194 a + 280567684548911174566145\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-422346227783811a+4481885041498204\right){x}-26439032262117056079194a+280567684548911174566145$ |
| 3.1-a2 |
3.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
3.1 |
\( 3 \) |
\( 3^{3} \) |
$2.37838$ |
$(11066a+106365)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1.829996857$ |
$24.61936384$ |
3.341617801 |
\( \frac{1790350}{27} a + \frac{17357831}{27} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 9 a - 96\) , \( -38 a + 403\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(9a-96\right){x}-38a+403$ |
| 3.2-a1 |
3.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( - 3^{6} \) |
$2.37838$ |
$(-11066a+117431)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.914998428$ |
$24.61936384$ |
3.341617801 |
\( -\frac{60146}{729} a + \frac{315247}{243} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 422346227783811 a + 4059538813714393\) , \( 26439032262117056079194 a + 254128652286794118486951\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(422346227783811a+4059538813714393\right){x}+26439032262117056079194a+254128652286794118486951$ |
| 3.2-a2 |
3.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
3.2 |
\( 3 \) |
\( 3^{3} \) |
$2.37838$ |
$(-11066a+117431)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 3 \) |
$1.829996857$ |
$24.61936384$ |
3.341617801 |
\( -\frac{1790350}{27} a + \frac{6382727}{9} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -9 a - 87\) , \( 38 a + 365\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-9a-87\right){x}+38a+365$ |
| 4.1-a1 |
4.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{13} \) |
$2.55573$ |
$(-219a+2324), (-219a-2105)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 11 \) |
$1$ |
$6.873619259$ |
7.477329157 |
\( \frac{6989311}{2048} a - \frac{36217713}{1024} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 50761281 a - 538672302\) , \( 633713937230 a - 6724892585789\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(50761281a-538672302\right){x}+633713937230a-6724892585789$ |
| 4.1-b1 |
4.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{13} \) |
$2.55573$ |
$(-219a+2324), (-219a-2105)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 11 \) |
$1$ |
$6.873619259$ |
7.477329157 |
\( -\frac{6989311}{2048} a - \frac{65446115}{2048} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -50761282 a - 487911020\) , \( -633713937231 a - 6091178648558\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-50761282a-487911020\right){x}-633713937231a-6091178648558$ |
| 6.1-a1 |
6.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3 \) |
$2.82838$ |
$(-219a+2324), (11066a+106365)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$5.731801885$ |
0.566838725 |
\( -\frac{69073}{12} a - \frac{747437}{12} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -24907258588263 a - 239405436418221\) , \( -216664947127985678648 a - 2082556217094143225097\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-24907258588263a-239405436418221\right){x}-216664947127985678648a-2082556217094143225097$ |
| 6.1-b1 |
6.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3^{3} \) |
$2.82838$ |
$(-219a+2324), (11066a+106365)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$11.11675565$ |
0.274844094 |
\( -\frac{2103789167}{110592} a + \frac{22594432109}{110592} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 10 a - 77\) , \( 93 a - 1017\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(10a-77\right){x}+93a-1017$ |
| 6.1-b2 |
6.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{6} \cdot 3^{6} \) |
$2.82838$ |
$(-219a+2324), (11066a+106365)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.558377826$ |
0.274844094 |
\( \frac{5582109053}{46656} a + \frac{53647896169}{46656} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 156401375870575 a - 1659711726708892\) , \( 12278710577422104844596 a - 130300132085004362603492\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(156401375870575a-1659711726708892\right){x}+12278710577422104844596a-130300132085004362603492$ |
| 6.1-c1 |
6.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3^{3} \) |
$2.82838$ |
$(-219a+2324), (11066a+106365)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \cdot 3 \) |
$1.119204095$ |
$10.06779306$ |
13.37190118 |
\( -\frac{2810617}{110592} a - \frac{143762741}{110592} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( -151421 a - 1455152\) , \( -149038252 a - 1432536013\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-151421a-1455152\right){x}-149038252a-1432536013$ |
| 6.4-a1 |
6.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3 \) |
$2.82838$ |
$(-219a-2105), (-11066a+117431)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$5.731801885$ |
0.566838725 |
\( \frac{69073}{12} a - \frac{136085}{2} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 24907258588261 a - 264312695006483\) , \( 216664947127985678647 a - 2299221164222128903745\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(24907258588261a-264312695006483\right){x}+216664947127985678647a-2299221164222128903745$ |
| 6.4-b1 |
6.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( - 2^{6} \cdot 3^{6} \) |
$2.82838$ |
$(-219a-2105), (-11066a+117431)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.558377826$ |
0.274844094 |
\( -\frac{5582109053}{46656} a + \frac{9871667537}{7776} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -156401375870575 a - 1503310350838317\) , \( -12278710577422104844596 a - 118021421507582257758896\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-156401375870575a-1503310350838317\right){x}-12278710577422104844596a-118021421507582257758896$ |
| 6.4-b2 |
6.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3^{3} \) |
$2.82838$ |
$(-219a-2105), (-11066a+117431)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$11.11675565$ |
0.274844094 |
\( \frac{2103789167}{110592} a + \frac{3415107157}{18432} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -10 a - 67\) , \( -93 a - 924\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-10a-67\right){x}-93a-924$ |
| 6.4-c1 |
6.4-c |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( 2^{12} \cdot 3^{3} \) |
$2.82838$ |
$(-219a-2105), (-11066a+117431)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \cdot 3 \) |
$1.119204095$ |
$10.06779306$ |
13.37190118 |
\( \frac{2810617}{110592} a - \frac{24428893}{18432} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( 151420 a - 1606572\) , \( 149038252 a - 1581574265\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(151420a-1606572\right){x}+149038252a-1581574265$ |
| 8.1-a1 |
8.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{9} \) |
$3.03929$ |
$(-219a+2324), (-219a-2105)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$12.35587858$ |
1.832876625 |
\( -\frac{4417}{2} a - 22721 \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -221588989960385556 a - 2129885497394360299\) , \( -183900163029282105030377285 a - 1767625233882558955565719462\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-221588989960385556a-2129885497394360299\right){x}-183900163029282105030377285a-1767625233882558955565719462$ |
| 8.1-a2 |
8.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{11} \) |
$3.03929$ |
$(-219a+2324), (-219a-2105)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$12.35587858$ |
1.832876625 |
\( \frac{167}{8} a + \frac{4471}{4} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -16 a + 261\) , \( -79 a + 1006\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-16a+261\right){x}-79a+1006$ |
| 8.1-b1 |
8.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{18} \) |
$3.03929$ |
$(-219a+2324), (-219a-2105)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.217879749$ |
$18.91318519$ |
3.416871804 |
\( \frac{161109}{1024} a + \frac{427653}{512} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -145477 a - 1397950\) , \( -54330839 a - 522219894\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-145477a-1397950\right){x}-54330839a-522219894$ |
| 8.1-b2 |
8.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{9} \) |
$3.03929$ |
$(-219a+2324), (-219a-2105)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$2.435759498$ |
$18.91318519$ |
3.416871804 |
\( \frac{1622025}{32} a + \frac{34017273}{16} \) |
\( \bigl[a\) , \( -a\) , \( 0\) , \( -48766729438261090337 a - 468739668899956594595\) , \( -589282423026818412281150467642 a - 5664108523165386864709634070782\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-48766729438261090337a-468739668899956594595\right){x}-589282423026818412281150467642a-5664108523165386864709634070782$ |
| 8.2-a1 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{9} \) |
$3.03929$ |
$(-219a+2324), (-219a-2105)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$12.35587858$ |
1.832876625 |
\( \frac{4417}{2} a - \frac{49859}{2} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 221588989960385556 a - 2351474487354745856\) , \( 183900163250871094990762841 a - 1951525399263315547950842603\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(221588989960385556a-2351474487354745856\right){x}+183900163250871094990762841a-1951525399263315547950842603$ |
| 8.2-a2 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{11} \) |
$3.03929$ |
$(-219a+2324), (-219a-2105)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$12.35587858$ |
1.832876625 |
\( -\frac{167}{8} a + \frac{9109}{8} \) |
\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 16 a + 244\) , \( 95 a + 1171\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(16a+244\right){x}+95a+1171$ |
| 8.2-b1 |
8.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{18} \) |
$3.03929$ |
$(-219a+2324), (-219a-2105)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1.217879749$ |
$18.91318519$ |
3.416871804 |
\( -\frac{161109}{1024} a + \frac{1016415}{1024} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 145477 a - 1543427\) , \( 54330839 a - 576550733\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(145477a-1543427\right){x}+54330839a-576550733$ |
| 8.2-b2 |
8.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{9} \) |
$3.03929$ |
$(-219a+2324), (-219a-2105)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 3 \) |
$2.435759498$ |
$18.91318519$ |
3.416871804 |
\( -\frac{1622025}{32} a + \frac{69656571}{32} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 48766729438261090337 a - 517506398338217684932\) , \( 589282423026818412281150467642 a - 6253390946192205276990784538424\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(48766729438261090337a-517506398338217684932\right){x}+589282423026818412281150467642a-6253390946192205276990784538424$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$3.13012$ |
$(11066a+106365), (-11066a+117431)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$4$ |
\( 1 \) |
$1$ |
$9.116790201$ |
1.803185050 |
\( \frac{1318912}{2187} a - \frac{13938688}{2187} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 11343802692046614214 a - 120379007209259130454\) , \( 70672049365122491986920576851 a - 749962897889782750578531864843\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(11343802692046614214a-120379007209259130454\right){x}+70672049365122491986920576851a-749962897889782750578531864843$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{6} \) |
$3.13012$ |
$(11066a+106365), (-11066a+117431)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.677182908$ |
$13.71764061$ |
2.755973298 |
\( \frac{68050117}{27} a + \frac{653860330}{27} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -6937 a - 66667\) , \( 916900 a + 8813181\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-6937a-66667\right){x}+916900a+8813181$ |
| 9.1-c1 |
9.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( - 3^{6} \) |
$3.13012$ |
$(11066a+106365), (-11066a+117431)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$0.677182908$ |
$13.71764061$ |
2.755973298 |
\( -\frac{68050117}{27} a + \frac{721910447}{27} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 6987 a - 73552\) , \( -983517 a + 10440155\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(6987a-73552\right){x}-983517a+10440155$ |
| 9.1-d1 |
9.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{8} \) |
$3.13012$ |
$(11066a+106365), (-11066a+117431)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$4$ |
\( 1 \) |
$1$ |
$9.116790201$ |
1.803185050 |
\( -\frac{1318912}{2187} a - \frac{4206592}{729} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -11343802692046614214 a - 109035204517212516240\) , \( -70672049365122491986920576851 a - 679290848524660258591611287992\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-11343802692046614214a-109035204517212516240\right){x}-70672049365122491986920576851a-679290848524660258591611287992$ |
| 9.2-a1 |
9.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{3} \) |
$3.13012$ |
$(11066a+106365)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$32.51285639$ |
3.215314562 |
\( 61 a + 489 \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 88691086789 a + 852487569798\) , \( 32991207397730746 a + 317107335479612277\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(88691086789a+852487569798\right){x}+32991207397730746a+317107335479612277$ |
| 9.2-b1 |
9.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{12} \) |
$3.13012$ |
$(11066a+106365)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.695227663$ |
$15.96284721$ |
2.676127058 |
\( \frac{60146}{729} a + \frac{885595}{729} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -5441647095476467 a - 52304427366474651\) , \( 289283375287006414744273 a + 2780555413747024648491188\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5441647095476467a-52304427366474651\right){x}+289283375287006414744273a+2780555413747024648491188$ |
| 9.2-b2 |
9.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{9} \) |
$3.13012$ |
$(11066a+106365)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.847613831$ |
$31.92569442$ |
2.676127058 |
\( \frac{1790350}{27} a + \frac{17357831}{27} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 81040673235413 a - 859993430112141\) , \( -1046916562007674859435 a + 11109746862379295274794\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(81040673235413a-859993430112141\right){x}-1046916562007674859435a+11109746862379295274794$ |
| 9.2-c1 |
9.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{9} \) |
$3.13012$ |
$(11066a+106365)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$15.86071708$ |
1.568523971 |
\( 61 a + 489 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 29 a + 258\) , \( 149 a + 1427\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(29a+258\right){x}+149a+1427$ |
| 9.3-a1 |
9.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{3} \) |
$3.13012$ |
$(-11066a+117431)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$32.51285639$ |
3.215314562 |
\( -61 a + 550 \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -88691086791 a + 941178656588\) , \( -32991207397730747 a + 350098542877343023\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-88691086791a+941178656588\right){x}-32991207397730747a+350098542877343023$ |
| 9.3-b1 |
9.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( - 3^{12} \) |
$3.13012$ |
$(-11066a+117431)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.695227663$ |
$15.96284721$ |
2.676127058 |
\( -\frac{60146}{729} a + \frac{315247}{243} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 5441647095476469 a - 57746074461951119\) , \( -289283369845359319267805 a + 3069838731287956601284342\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5441647095476469a-57746074461951119\right){x}-289283369845359319267805a+3069838731287956601284342$ |
| 9.3-b2 |
9.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{9} \) |
$3.13012$ |
$(-11066a+117431)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.847613831$ |
$31.92569442$ |
2.676127058 |
\( -\frac{1790350}{27} a + \frac{6382727}{9} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( -81040673235411 a - 778952756876729\) , \( 1046916480967001624023 a + 10062829521418863538630\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-81040673235411a-778952756876729\right){x}+1046916480967001624023a+10062829521418863538630$ |
| 9.3-c1 |
9.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{409}) \) |
$2$ |
$[2, 0]$ |
9.3 |
\( 3^{2} \) |
\( 3^{9} \) |
$3.13012$ |
$(-11066a+117431)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$15.86071708$ |
1.568523971 |
\( -61 a + 550 \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 22 a + 288\) , \( 109 a + 1168\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(22a+288\right){x}+109a+1168$ |
*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.
