Invariants
Level: | $12$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
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: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.24.0.24 |
Level structure
$\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&7\\2&5\end{bmatrix}$, $\begin{bmatrix}7&0\\2&11\end{bmatrix}$ |
$\GL_2(\Z/12\Z)$-subgroup: | $C_2^2:\GL(2,3)$ |
Contains $-I$: | no $\quad$ (see 12.12.0.d.1 for the level structure with $-I$) |
Cyclic 12-isogeny field degree: | $8$ |
Cyclic 12-torsion field degree: | $32$ |
Full 12-torsion field degree: | $192$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 64 x^{2} + 3 y^{2} + 3 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 |
---|---|---|---|---|---|
4.12.0-4.a.1.2 | $4$ | $2$ | $2$ | $0$ | $0$ |
12.12.0-4.a.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
12.72.2-12.o.1.4 | $12$ | $3$ | $3$ | $2$ |
12.96.1-12.g.1.1 | $12$ | $4$ | $4$ | $1$ |
36.648.22-36.s.1.2 | $36$ | $27$ | $27$ | $22$ |
60.120.4-60.h.1.3 | $60$ | $5$ | $5$ | $4$ |
60.144.3-60.di.1.8 | $60$ | $6$ | $6$ | $3$ |
60.240.7-60.p.1.4 | $60$ | $10$ | $10$ | $7$ |
84.192.5-84.h.1.10 | $84$ | $8$ | $8$ | $5$ |
84.504.16-84.p.1.8 | $84$ | $21$ | $21$ | $16$ |
132.288.9-132.h.1.10 | $132$ | $12$ | $12$ | $9$ |
156.336.11-156.l.1.3 | $156$ | $14$ | $14$ | $11$ |
204.432.15-204.l.1.2 | $204$ | $18$ | $18$ | $15$ |
228.480.17-228.h.1.10 | $228$ | $20$ | $20$ | $17$ |