Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | ||||
Index: | $24$ | $\PSL_2$-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 | $2\cdot4$ | ||
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: | 24.24.0.353 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&5\\14&5\end{bmatrix}$, $\begin{bmatrix}11&22\\8&3\end{bmatrix}$, $\begin{bmatrix}13&13\\12&23\end{bmatrix}$, $\begin{bmatrix}23&6\\4&19\end{bmatrix}$ |
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: | $3072$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 4 x^{2} + 4 x z + 6 y^{2} + 7 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 |
---|---|---|---|---|---|
8.12.0.q.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
24.12.0.bp.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.12.0.br.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 |
---|---|---|---|---|
24.48.1.gm.1 | $24$ | $2$ | $2$ | $1$ |
24.48.1.gn.1 | $24$ | $2$ | $2$ | $1$ |
24.48.1.gu.1 | $24$ | $2$ | $2$ | $1$ |
24.48.1.gv.1 | $24$ | $2$ | $2$ | $1$ |
24.48.1.ke.1 | $24$ | $2$ | $2$ | $1$ |
24.48.1.kf.1 | $24$ | $2$ | $2$ | $1$ |
24.48.1.km.1 | $24$ | $2$ | $2$ | $1$ |
24.48.1.kn.1 | $24$ | $2$ | $2$ | $1$ |
24.72.4.jk.1 | $24$ | $3$ | $3$ | $4$ |
24.96.3.hg.1 | $24$ | $4$ | $4$ | $3$ |
120.48.1.bby.1 | $120$ | $2$ | $2$ | $1$ |
120.48.1.bbz.1 | $120$ | $2$ | $2$ | $1$ |
120.48.1.bcc.1 | $120$ | $2$ | $2$ | $1$ |
120.48.1.bcd.1 | $120$ | $2$ | $2$ | $1$ |
120.48.1.bms.1 | $120$ | $2$ | $2$ | $1$ |
120.48.1.bmt.1 | $120$ | $2$ | $2$ | $1$ |
120.48.1.bmw.1 | $120$ | $2$ | $2$ | $1$ |
120.48.1.bmx.1 | $120$ | $2$ | $2$ | $1$ |
120.120.8.gw.1 | $120$ | $5$ | $5$ | $8$ |
120.144.7.gtf.1 | $120$ | $6$ | $6$ | $7$ |
120.240.15.bba.1 | $120$ | $10$ | $10$ | $15$ |
168.48.1.bbw.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bbx.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bca.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bcb.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bmq.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bmr.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bmu.1 | $168$ | $2$ | $2$ | $1$ |
168.48.1.bmv.1 | $168$ | $2$ | $2$ | $1$ |
168.192.11.pu.1 | $168$ | $8$ | $8$ | $11$ |
264.48.1.bbw.1 | $264$ | $2$ | $2$ | $1$ |
264.48.1.bbx.1 | $264$ | $2$ | $2$ | $1$ |
264.48.1.bca.1 | $264$ | $2$ | $2$ | $1$ |
264.48.1.bcb.1 | $264$ | $2$ | $2$ | $1$ |
264.48.1.bmq.1 | $264$ | $2$ | $2$ | $1$ |
264.48.1.bmr.1 | $264$ | $2$ | $2$ | $1$ |
264.48.1.bmu.1 | $264$ | $2$ | $2$ | $1$ |
264.48.1.bmv.1 | $264$ | $2$ | $2$ | $1$ |
264.288.19.bft.1 | $264$ | $12$ | $12$ | $19$ |
312.48.1.bby.1 | $312$ | $2$ | $2$ | $1$ |
312.48.1.bbz.1 | $312$ | $2$ | $2$ | $1$ |
312.48.1.bcc.1 | $312$ | $2$ | $2$ | $1$ |
312.48.1.bcd.1 | $312$ | $2$ | $2$ | $1$ |
312.48.1.bms.1 | $312$ | $2$ | $2$ | $1$ |
312.48.1.bmt.1 | $312$ | $2$ | $2$ | $1$ |
312.48.1.bmw.1 | $312$ | $2$ | $2$ | $1$ |
312.48.1.bmx.1 | $312$ | $2$ | $2$ | $1$ |
312.336.23.pz.1 | $312$ | $14$ | $14$ | $23$ |