Invariants
Level: | $88$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (all of which are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8C0 |
Level structure
$\GL_2(\Z/88\Z)$-generators: | $\begin{bmatrix}28&35\\49&14\end{bmatrix}$, $\begin{bmatrix}43&8\\32&67\end{bmatrix}$, $\begin{bmatrix}44&45\\7&2\end{bmatrix}$, $\begin{bmatrix}68&13\\27&62\end{bmatrix}$, $\begin{bmatrix}87&14\\64&37\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.12.0.n.1 for the level structure with $-I$) |
Cyclic 88-isogeny field degree: | $12$ |
Cyclic 88-torsion field degree: | $480$ |
Full 88-torsion field degree: | $844800$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
44.12.0-4.c.1.1 | $44$ | $2$ | $2$ | $0$ | $0$ |
88.12.0-4.c.1.3 | $88$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
88.48.0-8.i.1.1 | $88$ | $2$ | $2$ | $0$ |
88.48.0-8.k.1.6 | $88$ | $2$ | $2$ | $0$ |
88.48.0-8.q.1.1 | $88$ | $2$ | $2$ | $0$ |
88.48.0-8.r.1.6 | $88$ | $2$ | $2$ | $0$ |
88.48.0-8.ba.1.2 | $88$ | $2$ | $2$ | $0$ |
88.48.0-8.ba.2.5 | $88$ | $2$ | $2$ | $0$ |
88.48.0-8.bb.1.1 | $88$ | $2$ | $2$ | $0$ |
88.48.0-8.bb.2.6 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.bf.1.11 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.bh.1.6 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.bj.1.5 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.bl.1.8 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.bu.1.8 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.bu.2.4 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.bv.1.8 | $88$ | $2$ | $2$ | $0$ |
88.48.0-88.bv.2.6 | $88$ | $2$ | $2$ | $0$ |
88.288.9-88.bl.1.12 | $88$ | $12$ | $12$ | $9$ |
176.48.0-16.e.1.13 | $176$ | $2$ | $2$ | $0$ |
176.48.0-16.e.2.16 | $176$ | $2$ | $2$ | $0$ |
176.48.0-176.e.1.16 | $176$ | $2$ | $2$ | $0$ |
176.48.0-176.e.2.8 | $176$ | $2$ | $2$ | $0$ |
176.48.0-16.f.1.15 | $176$ | $2$ | $2$ | $0$ |
176.48.0-16.f.2.15 | $176$ | $2$ | $2$ | $0$ |
176.48.0-176.f.1.16 | $176$ | $2$ | $2$ | $0$ |
176.48.0-176.f.2.12 | $176$ | $2$ | $2$ | $0$ |
176.48.0-16.g.1.11 | $176$ | $2$ | $2$ | $0$ |
176.48.0-176.g.1.6 | $176$ | $2$ | $2$ | $0$ |
176.48.0-16.h.1.11 | $176$ | $2$ | $2$ | $0$ |
176.48.0-176.h.1.12 | $176$ | $2$ | $2$ | $0$ |
176.48.1-16.a.1.6 | $176$ | $2$ | $2$ | $1$ |
176.48.1-176.a.1.21 | $176$ | $2$ | $2$ | $1$ |
176.48.1-16.b.1.6 | $176$ | $2$ | $2$ | $1$ |
176.48.1-176.b.1.27 | $176$ | $2$ | $2$ | $1$ |
264.48.0-24.bh.1.2 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bj.1.8 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bl.1.2 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bn.1.8 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.by.1.3 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.by.2.3 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bz.1.3 | $264$ | $2$ | $2$ | $0$ |
264.48.0-24.bz.2.3 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.cx.1.17 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.cz.1.7 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.db.1.4 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.dd.1.13 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.ec.1.26 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.ec.2.20 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.ed.1.26 | $264$ | $2$ | $2$ | $0$ |
264.48.0-264.ed.2.20 | $264$ | $2$ | $2$ | $0$ |
264.72.2-24.cj.1.41 | $264$ | $3$ | $3$ | $2$ |
264.96.1-24.ir.1.39 | $264$ | $4$ | $4$ | $1$ |