Invariants
Level: | $8$ | $\SL_2$-level: | $8$ | ||||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.169 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&6\\4&3\end{bmatrix}$, $\begin{bmatrix}5&1\\0&1\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: | $D_4:C_4$ |
Contains $-I$: | no $\quad$ (see 8.24.0.j.1 for the level structure with $-I$) |
Cyclic 8-isogeny field degree: | $2$ |
Cyclic 8-torsion field degree: | $4$ |
Full 8-torsion field degree: | $32$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 16 x^{2} + y^{2} + z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.24.0-8.d.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-8.d.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-8.m.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-8.m.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-8.o.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-8.o.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
16.96.1-16.c.1.2 | $16$ | $2$ | $2$ | $1$ |
16.96.1-16.e.1.3 | $16$ | $2$ | $2$ | $1$ |
24.144.4-24.ci.1.14 | $24$ | $3$ | $3$ | $4$ |
24.192.3-24.co.1.8 | $24$ | $4$ | $4$ | $3$ |
40.240.8-40.w.1.8 | $40$ | $5$ | $5$ | $8$ |
40.288.7-40.bs.1.11 | $40$ | $6$ | $6$ | $7$ |
40.480.15-40.ci.1.10 | $40$ | $10$ | $10$ | $15$ |
48.96.1-48.c.1.4 | $48$ | $2$ | $2$ | $1$ |
48.96.1-48.e.1.4 | $48$ | $2$ | $2$ | $1$ |
56.384.11-56.bq.1.10 | $56$ | $8$ | $8$ | $11$ |
56.1008.34-56.ci.1.7 | $56$ | $21$ | $21$ | $34$ |
56.1344.45-56.ck.1.15 | $56$ | $28$ | $28$ | $45$ |
80.96.1-80.c.1.4 | $80$ | $2$ | $2$ | $1$ |
80.96.1-80.e.1.6 | $80$ | $2$ | $2$ | $1$ |
112.96.1-112.c.1.6 | $112$ | $2$ | $2$ | $1$ |
112.96.1-112.e.1.6 | $112$ | $2$ | $2$ | $1$ |
176.96.1-176.c.1.6 | $176$ | $2$ | $2$ | $1$ |
176.96.1-176.e.1.6 | $176$ | $2$ | $2$ | $1$ |
208.96.1-208.c.1.6 | $208$ | $2$ | $2$ | $1$ |
208.96.1-208.e.1.6 | $208$ | $2$ | $2$ | $1$ |
240.96.1-240.c.1.12 | $240$ | $2$ | $2$ | $1$ |
240.96.1-240.e.1.8 | $240$ | $2$ | $2$ | $1$ |
272.96.1-272.c.1.7 | $272$ | $2$ | $2$ | $1$ |
272.96.1-272.e.1.7 | $272$ | $2$ | $2$ | $1$ |
304.96.1-304.c.1.6 | $304$ | $2$ | $2$ | $1$ |
304.96.1-304.e.1.6 | $304$ | $2$ | $2$ | $1$ |