Invariants
| Level: | $8$ | $\SL_2$-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: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.192.1.34 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&2\\4&7\end{bmatrix}$, $\begin{bmatrix}7&2\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $D_4$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.c.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $8$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 2 y^{2} + w^{2} $ |
| $=$ | $2 y^{2} + 2 z^{2} - w^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(16z^{8}-32z^{6}w^{2}+20z^{4}w^{4}-4z^{2}w^{6}+w^{8})^{3}}{w^{8}z^{4}(z-w)^{2}(z+w)^{2}(2z^{2}-w^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.d.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.d.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.e.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.e.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.g.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.g.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.h.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.h.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.e.2.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.e.2.6 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.i.2.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.i.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.j.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.j.1.4 | $8$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 16.384.9-16.bb.2.1 | $16$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 16.384.9-16.bc.2.1 | $16$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.ks.2.3 | $24$ | $3$ | $3$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.dn.2.1 | $24$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 40.960.33-40.by.2.2 | $40$ | $5$ | $5$ | $33$ | $11$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.he.2.5 | $40$ | $6$ | $6$ | $33$ | $5$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.je.1.3 | $40$ | $10$ | $10$ | $65$ | $17$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 48.384.9-48.dh.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.dk.2.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 56.1536.49-56.dn.2.1 | $56$ | $8$ | $8$ | $49$ | $5$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.4032.145-56.kw.2.5 | $56$ | $21$ | $21$ | $145$ | $27$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.5376.193-56.lq.2.3 | $56$ | $28$ | $28$ | $193$ | $32$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 80.384.9-80.gf.2.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.gi.2.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.dh.2.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.dk.2.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.dh.2.1 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.dk.2.1 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.gf.2.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.gi.2.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.rv.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.se.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.gf.2.3 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.gi.2.3 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.dh.2.1 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.384.9-304.dk.2.1 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |