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 | $4^{6}$ | 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: | 4G0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.0.151 |
Level structure
$\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&6\\0&5\end{bmatrix}$, $\begin{bmatrix}3&6\\4&7\end{bmatrix}$, $\begin{bmatrix}7&6\\6&5\end{bmatrix}$ |
$\GL_2(\Z/8\Z)$-subgroup: | $C_2^3:C_4$ |
Contains $-I$: | no $\quad$ (see 8.24.0.b.1 for the level structure with $-I$) |
Cyclic 8-isogeny field degree: | $4$ |
Cyclic 8-torsion field degree: | $8$ |
Full 8-torsion field degree: | $32$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ x^{2} + 2 y^{2} + 16 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 |
---|---|---|---|---|---|
8.24.0-4.a.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
8.24.0-4.a.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
8.96.1-8.f.1.2 | $8$ | $2$ | $2$ | $1$ |
8.96.1-8.o.1.1 | $8$ | $2$ | $2$ | $1$ |
24.96.1-24.t.1.1 | $24$ | $2$ | $2$ | $1$ |
24.96.1-24.z.1.2 | $24$ | $2$ | $2$ | $1$ |
24.144.4-24.b.1.3 | $24$ | $3$ | $3$ | $4$ |
24.192.3-24.e.1.13 | $24$ | $4$ | $4$ | $3$ |
40.96.1-40.t.1.3 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.z.1.1 | $40$ | $2$ | $2$ | $1$ |
40.240.8-40.b.1.5 | $40$ | $5$ | $5$ | $8$ |
40.288.7-40.b.1.9 | $40$ | $6$ | $6$ | $7$ |
40.480.15-40.b.1.9 | $40$ | $10$ | $10$ | $15$ |
56.96.1-56.t.1.1 | $56$ | $2$ | $2$ | $1$ |
56.96.1-56.z.1.3 | $56$ | $2$ | $2$ | $1$ |
56.384.11-56.b.1.9 | $56$ | $8$ | $8$ | $11$ |
56.1008.34-56.b.1.10 | $56$ | $21$ | $21$ | $34$ |
56.1344.45-56.b.1.6 | $56$ | $28$ | $28$ | $45$ |
88.96.1-88.t.1.1 | $88$ | $2$ | $2$ | $1$ |
88.96.1-88.z.1.3 | $88$ | $2$ | $2$ | $1$ |
104.96.1-104.t.1.3 | $104$ | $2$ | $2$ | $1$ |
104.96.1-104.z.1.1 | $104$ | $2$ | $2$ | $1$ |
120.96.1-120.t.1.3 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.z.1.2 | $120$ | $2$ | $2$ | $1$ |
136.96.1-136.t.1.3 | $136$ | $2$ | $2$ | $1$ |
136.96.1-136.z.1.1 | $136$ | $2$ | $2$ | $1$ |
152.96.1-152.t.1.1 | $152$ | $2$ | $2$ | $1$ |
152.96.1-152.z.1.3 | $152$ | $2$ | $2$ | $1$ |
168.96.1-168.t.1.6 | $168$ | $2$ | $2$ | $1$ |
168.96.1-168.z.1.3 | $168$ | $2$ | $2$ | $1$ |
184.96.1-184.t.1.1 | $184$ | $2$ | $2$ | $1$ |
184.96.1-184.z.1.3 | $184$ | $2$ | $2$ | $1$ |
232.96.1-232.t.1.3 | $232$ | $2$ | $2$ | $1$ |
232.96.1-232.z.1.1 | $232$ | $2$ | $2$ | $1$ |
248.96.1-248.t.1.1 | $248$ | $2$ | $2$ | $1$ |
248.96.1-248.z.1.3 | $248$ | $2$ | $2$ | $1$ |
264.96.1-264.t.1.3 | $264$ | $2$ | $2$ | $1$ |
264.96.1-264.z.1.2 | $264$ | $2$ | $2$ | $1$ |
280.96.1-280.t.1.7 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.z.1.7 | $280$ | $2$ | $2$ | $1$ |
296.96.1-296.t.1.3 | $296$ | $2$ | $2$ | $1$ |
296.96.1-296.z.1.1 | $296$ | $2$ | $2$ | $1$ |
312.96.1-312.t.1.6 | $312$ | $2$ | $2$ | $1$ |
312.96.1-312.z.1.3 | $312$ | $2$ | $2$ | $1$ |
328.96.1-328.t.1.3 | $328$ | $2$ | $2$ | $1$ |
328.96.1-328.z.1.1 | $328$ | $2$ | $2$ | $1$ |