Invariants
| Level: | $88$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
Level structure
| $\GL_2(\Z/88\Z)$-generators: | $\begin{bmatrix}27&48\\48&87\end{bmatrix}$, $\begin{bmatrix}35&72\\44&57\end{bmatrix}$, $\begin{bmatrix}49&64\\52&57\end{bmatrix}$, $\begin{bmatrix}63&32\\20&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.g.2 for the level structure with $-I$) |
| Cyclic 88-isogeny field degree: | $12$ |
| Cyclic 88-torsion field degree: | $240$ |
| Full 88-torsion field degree: | $105600$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{708x^{2}y^{28}z^{2}-477870x^{2}y^{24}z^{6}+39545991x^{2}y^{20}z^{10}+258743115x^{2}y^{16}z^{14}+1130374344x^{2}y^{12}z^{18}+2157968565x^{2}y^{8}z^{22}+754974741x^{2}y^{4}z^{26}+16777215x^{2}z^{30}-8xy^{30}z+161586xy^{26}z^{5}-643608xy^{22}z^{9}+160418057xy^{18}z^{13}+1015012944xy^{14}z^{17}+2422211385xy^{10}z^{21}+2013265900xy^{6}z^{25}+184549377xy^{2}z^{29}+y^{32}-4152y^{28}z^{4}+12539228y^{24}z^{8}+76473134y^{20}z^{12}+457376604y^{16}z^{16}+1203755024y^{12}z^{20}+1409287006y^{8}z^{24}+167772138y^{4}z^{28}+z^{32}}{z^{2}y^{8}(x^{2}y^{20}+157x^{2}y^{16}z^{4}-16620x^{2}y^{12}z^{8}+311305x^{2}y^{8}z^{12}+983053x^{2}y^{4}z^{16}+65535x^{2}z^{20}-49xy^{18}z^{3}+200xy^{14}z^{7}+16365xy^{10}z^{11}+1310708xy^{6}z^{15}+458753xy^{2}z^{19}-6y^{20}z^{2}+1222y^{16}z^{6}-32504y^{12}z^{10}+655362y^{8}z^{14}+393202y^{4}z^{18}+z^{22})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 88.96.0-8.b.2.3 | $88$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 88.96.0-8.b.2.5 | $88$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 88.96.0-8.c.1.7 | $88$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 88.96.0-8.c.1.9 | $88$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 88.96.0-8.k.2.4 | $88$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 88.96.0-8.k.2.5 | $88$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 88.96.0-8.l.2.4 | $88$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 88.96.0-8.l.2.5 | $88$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 88.96.1-8.h.1.2 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-8.h.1.5 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-8.i.1.2 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-8.i.1.3 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-8.k.1.2 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.96.1-8.k.1.3 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 88.384.5-8.d.1.3 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.384.5-8.d.3.2 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.384.5-88.bb.1.5 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.384.5-88.bb.3.5 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.a.2.3 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.d.2.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.i.2.8 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.k.3.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.k.4.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.k.5.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.l.4.3 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.l.5.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.l.6.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.l.2.8 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.m.3.6 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.m.4.6 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.u.2.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-16.x.2.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bn.3.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bn.4.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bn.6.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bo.4.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bo.5.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bo.6.6 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bp.3.8 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bp.4.8 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.cw.2.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.cz.2.10 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.9-16.bq.1.6 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-16.bq.2.6 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.ga.1.14 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.ga.2.14 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.5-24.bj.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-24.bj.3.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.hv.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.hv.3.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |