| Curve name |
$X_{239c}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 29 & 29 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 29 & 29 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{239}$ |
| Curves that $X_{239c}$ minimally covers |
|
| Curves that minimally cover $X_{239c}$ |
|
| Curves that minimally cover $X_{239c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -756t^{32} - 13824t^{31} - 38016t^{30} + 1257984t^{29} + 16039296t^{28}
+ 66963456t^{27} - 231565824t^{26} - 4636624896t^{25} - 30197041920t^{24} -
121783246848t^{23} - 346143836160t^{22} - 760381267968t^{21} -
1500984317952t^{20} - 3151317270528t^{19} - 6790311419904t^{18} -
12378319257600t^{17} - 16212489320448t^{16} - 12120808882176t^{15} +
1420736495616t^{14} + 16306891259904t^{13} + 21383523237888t^{12} +
14311390445568t^{11} + 3081581494272t^{10} - 3899081097216t^{9} -
5066891919360t^{8} - 3466636296192t^{7} - 1762125152256t^{6} - 734458281984t^{5}
- 249580486656t^{4} - 64097353728t^{3} - 11154751488t^{2} - 1132462080t -
49545216\]
\[B(t) = 7344t^{48} + 228096t^{47} + 1783296t^{46} - 19229184t^{45} -
462309120t^{44} - 2975616000t^{43} + 6666541056t^{42} + 252556683264t^{41} +
1988117042688t^{40} + 6747626557440t^{39} - 16810441629696t^{38} -
382441504333824t^{37} - 2872083213103104t^{36} - 14665967556968448t^{35} -
56535279022080000t^{34} - 166055265929428992t^{33} - 349914212481871872t^{32} -
392427624079687680t^{31} + 505440014385807360t^{30} + 3915677458134466560t^{29}
+ 11380046589762600960t^{28} + 22176780819953614848t^{27} +
31246634414150516736t^{26} + 30704055260021784576t^{25} +
16008283247955148800t^{24} - 8421764080550805504t^{23} -
31313010525487497216t^{22} - 43298557511482736640t^{21} -
42452562827412504576t^{20} - 31383758135151820800t^{19} -
12899349951808536576t^{18} + 8623901100661014528t^{17} +
24973850881079377920t^{16} + 28556556538522632192t^{15} +
19607238490120519680t^{14} + 6227443568747741184t^{13} -
3160628383658803200t^{12} - 5923598974517772288t^{11} -
4513493882461224960t^{10} - 2258428448106086400t^{9} - 789879246683111424t^{8} -
183318721572372480t^{7} - 20007071671910400t^{6} + 3386427630944256t^{5} +
2020058679214080t^{4} + 447070662033408t^{3} + 56532507033600t^{2} +
4000762036224t + 123211874304\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 9x + 7$, with conductor $128$ |
| Generic density of odd order reductions |
$410657/1835008$ |