| 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 |
| 4.1-a1 |
4.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{36} \) |
$2.70154$ |
$(6987a-78176), (6987a+71189)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$4.088454276$ |
0.191249824 |
\( -\frac{1476800523}{134217728} a + \frac{352492037}{134217728} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -1797627 a - 18315624\) , \( 16799647005 a + 171167893354\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-1797627a-18315624\right){x}+16799647005a+171167893354$ |
| 4.1-a2 |
4.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{36} \) |
$2.70154$ |
$(6987a-78176), (6987a+71189)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$4.088454276$ |
0.191249824 |
\( \frac{1476800523}{134217728} a - \frac{562154243}{67108864} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 1797627 a - 20113251\) , \( -16799647005 a + 187967540359\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(1797627a-20113251\right){x}-16799647005a+187967540359$ |
| 4.1-b1 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{2} \) |
$2.70154$ |
$(6987a-78176), (6987a+71189)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$0.360371982$ |
0.421437257 |
\( -\frac{1728432036001}{2} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 163344816719010359726568 a - 1827629081724995545396078\) , \( 117485751970104030410777438349540696 a - 1314522133617860166671991700477400307\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(163344816719010359726568a-1827629081724995545396078\right){x}+117485751970104030410777438349540696a-1314522133617860166671991700477400307$ |
| 4.1-b2 |
4.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{10} \) |
$2.70154$ |
$(6987a-78176), (6987a+71189)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5^{2} \) |
$1$ |
$9.009299559$ |
0.421437257 |
\( -\frac{1}{32} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -13610933815432910568 a - 138678798850594549260\) , \( -21014053426409282188052722856184 a - 214107549685706159706062316713267\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-13610933815432910568a-138678798850594549260\right){x}-21014053426409282188052722856184a-214107549685706159706062316713267$ |
| 6.1-a1 |
6.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.98975$ |
$(6987a-78176), (-196a+2193)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.664791421$ |
$18.03994894$ |
3.092619331 |
\( -\frac{163663}{768} a + \frac{1793665}{768} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 676802 a - 7572545\) , \( -780300984 a + 8730615446\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(676802a-7572545\right){x}-780300984a+8730615446$ |
| 6.1-a2 |
6.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{2} \) |
$2.98975$ |
$(6987a-78176), (-196a+2193)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.832395710$ |
$18.03994894$ |
3.092619331 |
\( \frac{1399157}{144} a + \frac{14336917}{144} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -10075393175452795 a - 102655956047261842\) , \( -1792301473526103661546294 a - 18261363907664053540688320\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-10075393175452795a-102655956047261842\right){x}-1792301473526103661546294a-18261363907664053540688320$ |
| 6.2-a1 |
6.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{5} \cdot 3^{10} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5^{2} \) |
$1$ |
$13.57707209$ |
1.270217289 |
\( \frac{3279319769}{1889568} a + \frac{6754147171}{629856} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( 3587943960335842271 a - 40144712622192276658\) , \( -13513133943106243773596288339 a + 151195471492373667736309882359\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(3587943960335842271a-40144712622192276658\right){x}-13513133943106243773596288339a+151195471492373667736309882359$ |
| 6.2-a2 |
6.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$0.543082883$ |
1.270217289 |
\( \frac{18465221703630062418761}{18} a + \frac{62712688712703122929459}{6} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( 2442442704114693711 a - 27327952035147365828\) , \( 1027949969929204765000419207799 a - 11501505204372374415692922873225\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2442442704114693711a-27327952035147365828\right){x}+1027949969929204765000419207799a-11501505204372374415692922873225$ |
| 6.2-b1 |
6.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{2} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.583077120$ |
$27.90133659$ |
3.044057840 |
\( -\frac{2405}{144} a + \frac{77993}{48} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 107427 a + 1094588\) , \( 28549964 a + 290889216\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(107427a+1094588\right){x}+28549964a+290889216$ |
| 6.2-b2 |
6.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3 \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.166154241$ |
$55.80267319$ |
3.044057840 |
\( -\frac{312169}{12} a + \frac{1172529}{4} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 2388205541028 a - 26721104394893\) , \( -6557857313933730983 a + 73374417269485212433\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(2388205541028a-26721104394893\right){x}-6557857313933730983a+73374417269485212433$ |
| 6.2-c1 |
6.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{18} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$12.85283406$ |
1.202460436 |
\( \frac{528981570977}{774840978} a + \frac{1122323241595}{258280326} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -4927 a - 50084\) , \( 509059 a + 5187064\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-4927a-50084\right){x}+509059a+5187064$ |
| 6.2-c2 |
6.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{6} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$12.85283406$ |
1.202460436 |
\( \frac{385561152854921}{5832} a + \frac{1309465815850483}{1944} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -1491006262314766418 a - 15191533537697224277\) , \( 3240010085459028796892736025 a + 33011747247334563266914167241\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-1491006262314766418a-15191533537697224277\right){x}+3240010085459028796892736025a+33011747247334563266914167241$ |
| 6.2-d1 |
6.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{16} \cdot 3^{2} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$25.43771997$ |
9.519410810 |
\( -\frac{36294675365}{589824} a + \frac{8361019673}{196608} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -369384295 a - 3763574965\) , \( 12652225152112 a + 128910727998211\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-369384295a-3763574965\right){x}+12652225152112a+128910727998211$ |
| 6.2-d2 |
6.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$25.43771997$ |
9.519410810 |
\( \frac{15025554724141}{768} a + \frac{51030686223679}{256} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 35988717254 a - 402669809690\) , \( -32343040763831766 a + 361879140573405721\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(35988717254a-402669809690\right){x}-32343040763831766a+361879140573405721$ |
| 6.2-e1 |
6.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$54.65512714$ |
5.113318023 |
\( \frac{50681}{18} a + \frac{167875}{6} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -5654053 a - 57607860\) , \( 23486739445 a + 239301201505\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5654053a-57607860\right){x}+23486739445a+239301201505$ |
| 6.2-f1 |
6.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{6} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.324745532$ |
$12.72735864$ |
2.320088835 |
\( \frac{1579457}{5832} a + \frac{5437915}{1944} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -99281034 a - 1011552363\) , \( -1477821550103 a - 15057197416241\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-99281034a-1011552363\right){x}-1477821550103a-15057197416241$ |
| 6.2-f2 |
6.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.974236598$ |
$12.72735864$ |
2.320088835 |
\( \frac{2830025}{18} a - \frac{4078157}{6} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 760891523942240568002 a - 8513447228552590474455\) , \( 37365919982458279207860700197514 a - 418079026914124626708411470111789\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(760891523942240568002a-8513447228552590474455\right){x}+37365919982458279207860700197514a-418079026914124626708411470111789$ |
| 6.2-g1 |
6.2-g |
$1$ |
$1$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{7} \cdot 3^{8} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$1.270638806$ |
$7.872571925$ |
13.10203283 |
\( \frac{42883939361}{839808} a + \frac{145727998459}{279936} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( -27227751259200422386 a - 277417544690968790852\) , \( -252639300917614073247819093993 a - 2574086044998836837225109848878\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-27227751259200422386a-277417544690968790852\right){x}-252639300917614073247819093993a-2574086044998836837225109848878$ |
| 6.2-h1 |
6.2-h |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{15} \cdot 3^{6} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$2.647204485$ |
0.247662005 |
\( \frac{311252873078873}{23887872} a - \frac{1160483950706717}{7962624} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 1007173847636 a - 11269045759749\) , \( 1798816917659029249 a - 20126565246743730767\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(1007173847636a-11269045759749\right){x}+1798816917659029249a-20126565246743730767$ |
| 6.2-h2 |
6.2-h |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{5} \cdot 3^{18} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$2.647204485$ |
0.247662005 |
\( \frac{37348177385969}{12397455648} a + \frac{126170000193259}{4132485216} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -38401432366532240535 a - 391263713933771222454\) , \( -417760201976105416924764390371 a - 4256466441115040080323818933150\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-38401432366532240535a-391263713933771222454\right){x}-417760201976105416924764390371a-4256466441115040080323818933150$ |
| 6.2-i1 |
6.2-i |
$1$ |
$1$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{13} \cdot 3^{2} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$12.72148910$ |
1.190172320 |
\( \frac{44514497}{73728} a + \frac{180562651}{24576} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( -37836076479016147 a - 385503427647245515\) , \( -12268801568047814143313819 a - 125004109774157050479475063\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-37836076479016147a-385503427647245515\right){x}-12268801568047814143313819a-125004109774157050479475063$ |
| 6.2-j1 |
6.2-j |
$1$ |
$1$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{4} \) |
$2.98975$ |
$(6987a-78176), (-196a-1997)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.367574895$ |
$10.22897970$ |
1.407051673 |
\( \frac{3201947393}{41472} a + \frac{10707210139}{13824} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -10791 a - 109794\) , \( 1864040 a + 18992640\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-10791a-109794\right){x}+1864040a+18992640$ |
| 6.3-a1 |
6.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{5} \cdot 3^{10} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5^{2} \) |
$1$ |
$13.57707209$ |
1.270217289 |
\( -\frac{3279319769}{1889568} a + \frac{11770880641}{944784} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -3587943960335842272 a - 36556768661856434386\) , \( 13513133943106243773596288339 a + 137682337549267423962713594020\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-3587943960335842272a-36556768661856434386\right){x}+13513133943106243773596288339a+137682337549267423962713594020$ |
| 6.3-a2 |
6.3-a |
$2$ |
$5$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$0.543082883$ |
1.270217289 |
\( -\frac{18465221703630062418761}{18} a + \frac{103301643920869715603569}{9} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -2442442704114693712 a - 24885509331032672116\) , \( -1027949969929204765000419207799 a - 10473555234443169650692503665426\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-2442442704114693712a-24885509331032672116\right){x}-1027949969929204765000419207799a-10473555234443169650692503665426$ |
| 6.3-b1 |
6.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{2} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.583077120$ |
$27.90133659$ |
3.044057840 |
\( \frac{2405}{144} a + \frac{115787}{72} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -107428 a + 1202015\) , \( -28549965 a + 319439180\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-107428a+1202015\right){x}-28549965a+319439180$ |
| 6.3-b2 |
6.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( 2^{2} \cdot 3 \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.166154241$ |
$55.80267319$ |
3.044057840 |
\( \frac{312169}{12} a + \frac{1602709}{6} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -2388205541029 a - 24332898853865\) , \( 6557857313933730982 a + 66816559955551481450\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2388205541029a-24332898853865\right){x}+6557857313933730982a+66816559955551481450$ |
| 6.3-c1 |
6.3-c |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{18} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$12.85283406$ |
1.202460436 |
\( -\frac{528981570977}{774840978} a + \frac{1947975647881}{387420489} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 4927 a - 55011\) , \( -513986 a + 5751134\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4927a-55011\right){x}-513986a+5751134$ |
| 6.3-c2 |
6.3-c |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{6} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$12.85283406$ |
1.202460436 |
\( -\frac{385561152854921}{5832} a + \frac{2156979300203185}{2916} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 1491006262314766418 a - 16682539800011990695\) , \( -3240010086950035059207502443 a + 36251757349476131863818893961\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1491006262314766418a-16682539800011990695\right){x}-3240010086950035059207502443a+36251757349476131863818893961$ |
| 6.3-d1 |
6.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$25.43771997$ |
9.519410810 |
\( -\frac{15025554724141}{768} a + \frac{84058806697589}{384} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -35988717254 a - 366681092436\) , \( 32343040763831766 a + 329536099809573955\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-35988717254a-366681092436\right){x}+32343040763831766a+329536099809573955$ |
| 6.3-d2 |
6.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{16} \cdot 3^{2} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$25.43771997$ |
9.519410810 |
\( \frac{36294675365}{589824} a - \frac{5605808173}{294912} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 369384295 a - 4132959260\) , \( -12652225152112 a + 141562953150323\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(369384295a-4132959260\right){x}-12652225152112a+141562953150323$ |
| 6.3-e1 |
6.3-e |
$1$ |
$1$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$54.65512714$ |
5.113318023 |
\( -\frac{50681}{18} a + \frac{277153}{9} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( 5654110 a - 63262026\) , \( -23544347362 a + 263432506184\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(5654110a-63262026\right){x}-23544347362a+263432506184$ |
| 6.3-f1 |
6.3-f |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2 \cdot 3^{2} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.974236598$ |
$12.72735864$ |
2.320088835 |
\( -\frac{2830025}{18} a - \frac{4702223}{9} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -760891523942240568002 a - 7752555704610349906567\) , \( -37365919981697387683918459629513 a - 380713106923913791795940420007822\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-760891523942240568002a-7752555704610349906567\right){x}-37365919981697387683918459629513a-380713106923913791795940420007822$ |
| 6.3-f2 |
6.3-f |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{3} \cdot 3^{6} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$0.324745532$ |
$12.72735864$ |
2.320088835 |
\( -\frac{1579457}{5832} a + \frac{8946601}{2916} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 99281034 a - 1110833511\) , \( 1477722269068 a - 16533908132947\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(99281034a-1110833511\right){x}+1477722269068a-16533908132947$ |
| 6.3-g1 |
6.3-g |
$1$ |
$1$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{7} \cdot 3^{8} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$1.270638806$ |
$7.872571925$ |
13.10203283 |
\( -\frac{42883939361}{839808} a + \frac{240033967369}{419904} \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 27227751259200422442 a - 304645295950169213352\) , \( 252639300612968777297649880697 a - 2826725342507841970973911574323\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(27227751259200422442a-304645295950169213352\right){x}+252639300612968777297649880697a-2826725342507841970973911574323$ |
| 6.3-h1 |
6.3-h |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{15} \cdot 3^{6} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$2.647204485$ |
0.247662005 |
\( -\frac{311252873078873}{23887872} a - \frac{1585099489520639}{11943936} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -1007173847636 a - 10261871912113\) , \( -1798815910485181613 a - 18327738067212789405\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1007173847636a-10261871912113\right){x}-1798815910485181613a-18327738067212789405$ |
| 6.3-h2 |
6.3-h |
$2$ |
$3$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{5} \cdot 3^{18} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1$ |
$2.647204485$ |
0.247662005 |
\( -\frac{37348177385969}{12397455648} a + \frac{207929088982873}{6198727824} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 38401432366532240535 a - 429665146300303462989\) , \( 417760201937703984558232149836 a - 4674226642661480350948279860532\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(38401432366532240535a-429665146300303462989\right){x}+417760201937703984558232149836a-4674226642661480350948279860532$ |
| 6.3-i1 |
6.3-i |
$1$ |
$1$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{13} \cdot 3^{2} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$12.72148910$ |
1.190172320 |
\( -\frac{44514497}{73728} a + \frac{293101225}{36864} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 37836076479016145 a - 423339504126261661\) , \( 12268801568047814143313818 a - 137272911342204864622788882\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(37836076479016145a-423339504126261661\right){x}+12268801568047814143313818a-137272911342204864622788882$ |
| 6.3-j1 |
6.3-j |
$1$ |
$1$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.3 |
\( 2 \cdot 3 \) |
\( - 2^{9} \cdot 3^{4} \) |
$2.98975$ |
$(6987a+71189), (-196a+2193)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \) |
$0.367574895$ |
$10.22897970$ |
1.407051673 |
\( -\frac{3201947393}{41472} a + \frac{17661788905}{20736} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( 10790 a - 120584\) , \( -1864040 a + 20856680\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(10790a-120584\right){x}-1864040a+20856680$ |
| 6.4-a1 |
6.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( - 2^{4} \cdot 3^{2} \) |
$2.98975$ |
$(6987a+71189), (-196a-1997)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.832395710$ |
$18.03994894$ |
3.092619331 |
\( -\frac{1399157}{144} a + \frac{2622679}{24} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 10075393175452795 a - 112731349222714637\) , \( 1792301473526103661546294 a - 20053665381190157202234614\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(10075393175452795a-112731349222714637\right){x}+1792301473526103661546294a-20053665381190157202234614$ |
| 6.4-a2 |
6.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{457}) \) |
$2$ |
$[2, 0]$ |
6.4 |
\( 2 \cdot 3 \) |
\( - 2^{8} \cdot 3 \) |
$2.98975$ |
$(6987a+71189), (-196a-1997)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$3.664791421$ |
$18.03994894$ |
3.092619331 |
\( \frac{163663}{768} a + \frac{271667}{128} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -676802 a - 6895743\) , \( 780300984 a + 7950314462\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-676802a-6895743\right){x}+780300984a+7950314462$ |
*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.
