Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8C1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.116 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&14\\18&11\end{bmatrix}$, $\begin{bmatrix}7&1\\12&17\end{bmatrix}$, $\begin{bmatrix}23&6\\20&7\end{bmatrix}$, $\begin{bmatrix}23&15\\12&13\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 48.48.1-24.bi.1.1, 48.48.1-24.bi.1.2, 48.48.1-24.bi.1.3, 48.48.1-24.bi.1.4, 48.48.1-24.bi.1.5, 48.48.1-24.bi.1.6, 48.48.1-24.bi.1.7, 48.48.1-24.bi.1.8, 240.48.1-24.bi.1.1, 240.48.1-24.bi.1.2, 240.48.1-24.bi.1.3, 240.48.1-24.bi.1.4, 240.48.1-24.bi.1.5, 240.48.1-24.bi.1.6, 240.48.1-24.bi.1.7, 240.48.1-24.bi.1.8 |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $3072$ |
Jacobian
Conductor: | $2^{6}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 6 x y - z w $ |
$=$ | $96 x^{2} + 6 y^{2} + 8 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 6 x^{4} + 18 x^{2} y^{2} + x^{2} z^{2} + 6 y^{2} z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}z$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{13440y^{2}z^{4}-12960y^{2}z^{2}w^{2}+2910y^{2}w^{4}+13824z^{6}-17984z^{4}w^{2}+6248z^{2}w^{4}-27w^{6}}{384y^{2}z^{4}+96y^{2}z^{2}w^{2}-6y^{2}w^{4}+512z^{6}+64z^{4}w^{2}+24z^{2}w^{4}-w^{6}}$ |
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 | Kernel decomposition |
---|---|---|---|---|---|---|
8.12.1.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.12.0.br.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.0.bs.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.48.1.h.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.cd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.ee.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.eh.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.gl.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.gn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.gy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.he.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.72.5.es.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
24.96.5.co.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
120.48.1.ro.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.rs.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.se.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.si.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.vg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.vk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.vw.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.wa.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.120.9.co.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
120.144.9.gdm.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
120.240.17.bma.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
168.48.1.rm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.rq.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.sc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.sg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.ve.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.vi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.vu.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.vy.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.13.cw.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
264.48.1.rm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.rq.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.sc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.sg.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.ve.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.vi.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.vu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.vy.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.288.21.cw.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
312.48.1.ro.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.rs.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.se.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.si.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.vg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.vk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.vw.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.wa.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |