| 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 |
| 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 |
\( \frac{3201947393}{41472} a + \frac{10707210139}{13824} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -10791 a - 109794\) , \( 1864040 a + 18992640\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-10791a-109794\right){x}+1864040a+18992640$ |
*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.
