Invariants
Level: | $24$ | $\SL_2$-level: | $4$ | ||||
Index: | $12$ | $\PSL_2$-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: | $1 \le \gamma \le 2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.12.0.63 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&5\\8&7\end{bmatrix}$, $\begin{bmatrix}11&12\\0&17\end{bmatrix}$, $\begin{bmatrix}17&0\\14&1\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $6144$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 25 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{3^2\cdot11^4}\cdot\frac{(x+3y)^{12}(263x^{4}-2080x^{3}y-8964x^{2}y^{2}+4160xy^{3}+1052y^{4})^{3}}{(x+3y)^{12}(x^{2}+2y^{2})^{4}(5x^{2}+52xy-10y^{2})^{2}}$ |
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 |
---|---|---|---|---|---|
8.6.0.a.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
12.6.0.f.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
24.6.0.d.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
24.36.2.u.1 | $24$ | $3$ | $3$ | $2$ |
24.48.1.cr.1 | $24$ | $4$ | $4$ | $1$ |
72.324.22.y.1 | $72$ | $27$ | $27$ | $22$ |
120.60.4.k.1 | $120$ | $5$ | $5$ | $4$ |
120.72.3.ka.1 | $120$ | $6$ | $6$ | $3$ |
120.120.7.w.1 | $120$ | $10$ | $10$ | $7$ |
168.96.5.be.1 | $168$ | $8$ | $8$ | $5$ |
168.252.16.w.1 | $168$ | $21$ | $21$ | $16$ |
168.336.21.w.1 | $168$ | $28$ | $28$ | $21$ |
264.144.9.bti.1 | $264$ | $12$ | $12$ | $9$ |
312.168.11.q.1 | $312$ | $14$ | $14$ | $11$ |