Invariants
Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $32$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $16$ 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: | 24H1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.79 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&17\\14&7\end{bmatrix}$, $\begin{bmatrix}11&4\\8&17\end{bmatrix}$, $\begin{bmatrix}17&10\\4&7\end{bmatrix}$, $\begin{bmatrix}17&15\\6&13\end{bmatrix}$, $\begin{bmatrix}23&4\\16&13\end{bmatrix}$ |
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: | $1024$ |
Jacobian
Conductor: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - y z $ |
$=$ | $4 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} + y^{2} z^{2} + z^{4} $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(3z^{6}-3z^{4}w^{2}-3z^{2}w^{4}-w^{6})^{3}}{z^{6}(z^{2}+w^{2})^{6}}$ |
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 |
---|---|---|---|---|---|---|
12.36.0.m.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.36.0.cg.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.36.1.gm.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.144.5.xy.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
24.144.5.xy.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
24.144.5.xz.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
24.144.5.xz.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
24.144.5.ya.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
24.144.5.ya.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
24.144.5.yb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
24.144.5.yb.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
24.144.9.ev.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.ns.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.wi.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.wt.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
24.144.9.dga.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.dgd.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
24.144.9.dgi.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
24.144.9.dgl.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
48.144.3.b.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
48.144.3.b.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
48.144.3.r.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
48.144.3.r.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
48.144.7.yw.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
48.144.7.yx.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
48.144.7.zi.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
48.144.7.zj.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
48.144.7.zm.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
48.144.7.zn.1 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
48.144.7.zq.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
48.144.7.zr.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
48.144.11.jt.1 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
48.144.11.jt.2 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
48.144.11.rr.1 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
48.144.11.rr.2 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
72.216.13.ng.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.144.5.kww.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.kww.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.kwx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.kwx.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.kwy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.kwy.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.kwz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.5.kwz.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
120.144.9.bgct.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgcv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgdj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgdl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgff.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgfh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgfv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.bgfx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.5.hwv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.hwv.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.hww.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.hww.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.hwx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.hwx.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.hwy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.5.hwy.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.144.9.bbzn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bbzp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcad.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcaf.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bcbz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bccb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bccp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.bccr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
240.144.3.bh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.144.3.bh.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.144.3.bx.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.144.3.bx.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.144.7.dwe.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
240.144.7.dwf.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
240.144.7.dwi.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
240.144.7.dwj.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
240.144.7.dwu.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
240.144.7.dwv.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
240.144.7.dwy.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
240.144.7.dwz.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
240.144.11.ctd.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
240.144.11.ctd.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
240.144.11.cub.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
240.144.11.cub.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
264.144.5.hww.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.144.5.hww.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.144.5.hwx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.144.5.hwx.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.144.5.hwy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.144.5.hwy.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.144.5.hwz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.144.5.hwz.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.144.9.bcfn.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcfp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcgd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcgf.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bchz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcib.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcip.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.bcir.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.5.hww.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.144.5.hww.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.144.5.hwx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.144.5.hwx.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.144.5.hwy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.144.5.hwy.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.144.5.hwz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.144.5.hwz.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.144.9.bbzv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bbzx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcal.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcan.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bcch.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bccj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bccx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.bccz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |