Invariants
Level: | $12$ | $\SL_2$-level: | $4$ | ||||
Index: | $8$ | $\PSL_2$-index: | $8$ | ||||
Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
Cusps: | $2$ (none of which are rational) | Cusp widths | $4^{2}$ | Cusp orbits | $2$ | ||
Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4D0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.8.0.8 |
Level structure
$\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}5&2\\5&3\end{bmatrix}$, $\begin{bmatrix}10&11\\7&6\end{bmatrix}$ |
$\GL_2(\Z/12\Z)$-subgroup: | $D_6:\GL(2,3)$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 12-isogeny field degree: | $24$ |
Cyclic 12-torsion field degree: | $96$ |
Full 12-torsion field degree: | $576$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 280 x^{2} - 16 x y + 16 x z + y^{2} + y z + z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
12.2.0.a.1 | $12$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
12.24.0.m.1 | $12$ | $3$ | $3$ | $0$ |
12.24.2.d.1 | $12$ | $3$ | $3$ | $2$ |
12.32.1.d.1 | $12$ | $4$ | $4$ | $1$ |
24.32.1.g.1 | $24$ | $4$ | $4$ | $1$ |
36.216.14.d.1 | $36$ | $27$ | $27$ | $14$ |
60.40.2.d.1 | $60$ | $5$ | $5$ | $2$ |
60.48.3.v.1 | $60$ | $6$ | $6$ | $3$ |
60.80.5.d.1 | $60$ | $10$ | $10$ | $5$ |
84.64.3.d.1 | $84$ | $8$ | $8$ | $3$ |
84.168.12.d.1 | $84$ | $21$ | $21$ | $12$ |
84.224.15.d.1 | $84$ | $28$ | $28$ | $15$ |
132.96.7.d.1 | $132$ | $12$ | $12$ | $7$ |
156.112.7.d.1 | $156$ | $14$ | $14$ | $7$ |
204.144.11.d.1 | $204$ | $18$ | $18$ | $11$ |
228.160.11.d.1 | $228$ | $20$ | $20$ | $11$ |
276.192.15.d.1 | $276$ | $24$ | $24$ | $15$ |