Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $72$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $6^{2}\cdot12^{3}\cdot24$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24G4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.144.4.103 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&4\\16&17\end{bmatrix}$, $\begin{bmatrix}3&20\\4&3\end{bmatrix}$, $\begin{bmatrix}5&0\\12&19\end{bmatrix}$, $\begin{bmatrix}9&10\\8&15\end{bmatrix}$, $\begin{bmatrix}11&22\\20&5\end{bmatrix}$, $\begin{bmatrix}15&10\\4&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.z.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $512$ |
Jacobian
| Conductor: | $2^{10}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}\cdot2$ |
| Newforms: | 36.2.a.a$^{2}$, 72.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 4 x y - z w $ |
| $=$ | $2 x^{3} - x z^{2} - 16 y^{3} + y w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{6} - 2 x^{4} z^{2} + x^{2} y^{3} z - 2 y^{3} z^{3} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1)$, $(0:1/4:0:1)$, $(0:-1/4:0:1)$, $(0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1920x^{2}z^{10}+1728x^{2}z^{7}w^{3}-3592x^{2}z^{4}w^{6}+1538x^{2}zw^{9}+9728y^{2}z^{9}w-13184y^{2}z^{6}w^{4}+18432y^{2}z^{3}w^{7}-1020y^{2}w^{10}+64z^{12}-512z^{9}w^{3}+3060z^{6}w^{6}-2431z^{3}w^{9}+64w^{12}}{w^{3}z^{6}(2x^{2}z-12y^{2}w-z^{3}+w^{3})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.z.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{6}+X^{2}Y^{3}Z-2X^{4}Z^{2}-2Y^{3}Z^{3} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.1.15 | $8$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.e.1.15 | $8$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.72.2-12.b.1.15 | $24$ | $2$ | $2$ | $2$ | $0$ | $2$ |
| 24.72.2-12.b.1.41 | $24$ | $2$ | $2$ | $2$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.288.7-24.ei.2.23 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.em.2.22 | $24$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2$ |
| 24.288.7-24.eq.2.21 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.eu.2.23 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.fa.2.46 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.fe.2.27 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.fi.2.27 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.fm.2.25 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.8-24.h.2.22 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.l.1.13 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.t.2.10 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.x.1.16 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.bd.2.6 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.bf.2.13 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.bl.2.15 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.bn.2.11 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.bt.2.19 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.bv.2.28 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.cb.2.20 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.cd.2.27 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.cj.2.28 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.cl.2.18 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.cr.2.25 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.ct.2.19 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.cz.1.17 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.db.1.19 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.dh.1.17 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.dj.1.19 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.dp.1.20 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.dr.1.21 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.dx.1.18 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.dz.1.17 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.eh.2.23 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.ej.2.29 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.ep.2.22 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.er.2.22 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.ez.2.31 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.fd.1.24 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.fl.2.23 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.fp.2.23 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.9-24.df.2.4 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.dn.2.10 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.ee.2.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 24.288.9-24.ei.1.10 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 24.288.9-24.gv.2.11 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.gx.2.3 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.hd.2.9 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.hf.2.3 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.jl.1.29 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.jn.1.17 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.jt.1.19 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 24.288.9-24.jv.1.25 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 24.288.9-24.kj.1.21 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.kl.1.31 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.kr.1.26 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.kt.1.20 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 72.432.16-72.br.2.59 | $72$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 120.288.7-120.bai.2.21 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.baq.2.53 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bay.2.54 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bbg.2.53 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bbo.2.59 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bbw.2.50 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bce.2.9 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcm.2.18 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.8-120.ch.2.27 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.cl.2.18 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.cx.2.10 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.db.2.24 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.dn.2.6 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.dr.2.28 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.ed.2.16 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.eh.2.10 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.fz.2.22 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.gd.2.53 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.gp.2.21 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.gt.2.54 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.hf.2.45 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.hj.2.12 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.hv.2.44 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.hz.2.10 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.il.2.36 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.ip.2.35 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.jb.2.36 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.jf.2.35 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.jr.2.35 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.jv.2.38 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.kh.2.33 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.kl.2.34 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.mf.2.30 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.mj.2.58 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.mv.2.27 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.mz.2.27 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.nl.2.61 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.np.2.31 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.ob.2.29 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.of.2.48 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.9-120.rz.2.7 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.sd.2.34 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.sp.2.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.st.2.25 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tf.2.49 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tj.2.6 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tv.2.18 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tz.2.17 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bcx.2.38 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bdb.2.56 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bdn.2.54 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bdr.2.40 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bed.2.56 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.beh.2.42 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bet.2.40 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bex.2.54 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.7-168.yw.2.61 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.ze.2.59 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.zm.2.62 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.zu.2.63 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bac.2.59 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bak.2.64 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bas.2.61 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bba.2.58 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.8-168.bz.2.25 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.cd.2.11 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.cp.2.11 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.ct.2.39 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.df.2.7 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.dj.2.26 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.dv.2.14 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.dz.2.7 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.fr.2.26 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.fv.2.57 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.gh.2.25 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.gl.2.58 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.gx.2.53 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.hb.2.20 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.hn.2.52 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.hr.2.18 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.id.2.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.ih.2.36 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.it.2.36 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.ix.2.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.jj.2.36 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.jn.2.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.jz.2.36 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.kd.2.34 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.lx.2.30 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.mb.2.58 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.mn.2.27 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.mr.2.27 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.nd.2.61 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.nh.2.31 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.nt.2.29 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.nx.2.48 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.9-168.rr.2.7 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.rv.2.34 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.sh.2.2 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.sl.2.41 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
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