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: | 24.144.4.9 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&8\\16&13\end{bmatrix}$, $\begin{bmatrix}9&14\\8&15\end{bmatrix}$, $\begin{bmatrix}15&14\\4&15\end{bmatrix}$, $\begin{bmatrix}21&4\\4&9\end{bmatrix}$, $\begin{bmatrix}21&14\\4&3\end{bmatrix}$, $\begin{bmatrix}23&22\\4&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.z.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| 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 $ | $=$ | $ 2 x z - y w $ |
| $=$ | $x^{3} - x y^{2} + 8 z^{3} - z w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{6} - 2 x^{4} y^{2} - x^{2} y z^{3} + 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 |
|---|
| $(0:0:0:1)$, $(0:1:0:0)$, $(1:1:0:0)$, $(-1:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{960x^{2}y^{10}-864x^{2}y^{7}w^{3}-1796x^{2}y^{4}w^{6}-769x^{2}yw^{9}+64y^{12}-4864y^{9}z^{2}w+512y^{9}w^{3}-6592y^{6}z^{2}w^{4}+3060y^{6}w^{6}-9216y^{3}z^{2}w^{7}+2431y^{3}w^{9}-510z^{2}w^{10}+64w^{12}}{w^{3}y^{6}(x^{2}y-y^{3}+6z^{2}w-w^{3})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.z.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{6}-2X^{4}Y^{2}-X^{2}YZ^{3}+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.2.13 | $8$ | $3$ | $3$ | $0$ | $0$ | 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.2.13 | $8$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 12.72.2-12.b.1.8 | $12$ | $2$ | $2$ | $2$ | $0$ | $2$ |
| 24.72.2-12.b.1.1 | $24$ | $2$ | $2$ | $2$ | $0$ | $2$ |
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.1.26 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.em.1.30 | $24$ | $2$ | $2$ | $7$ | $1$ | $1\cdot2$ |
| 24.288.7-24.eq.1.28 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.eu.1.26 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.fa.1.48 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.fe.1.30 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.fi.1.29 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.7-24.fm.1.30 | $24$ | $2$ | $2$ | $7$ | $0$ | $1\cdot2$ |
| 24.288.8-24.h.1.45 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.l.1.1 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.t.1.4 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.x.1.29 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.bd.1.2 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.bf.1.31 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.bl.1.29 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.bn.1.4 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.bt.1.20 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.bv.1.20 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.cb.1.20 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.cd.1.20 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.cj.1.19 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.cl.1.19 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.cr.1.20 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.ct.1.20 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.cz.2.25 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.db.2.29 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.dh.2.25 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.dj.2.27 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.dp.2.29 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.dr.2.25 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.dx.2.28 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.dz.2.29 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.eh.1.20 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.ej.1.30 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.ep.1.7 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.er.1.19 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.ez.1.30 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.fd.1.20 | $24$ | $2$ | $2$ | $8$ | $1$ | $1^{2}\cdot2$ |
| 24.288.8-24.fl.1.20 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.8-24.fp.1.12 | $24$ | $2$ | $2$ | $8$ | $0$ | $1^{2}\cdot2$ |
| 24.288.9-24.df.1.19 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.dn.1.7 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.ee.1.30 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 24.288.9-24.ei.1.20 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 24.288.9-24.gv.1.32 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.gx.1.20 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.hd.1.20 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.hf.1.30 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.jl.2.30 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.jn.2.25 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.jt.2.29 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 24.288.9-24.jv.2.26 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{3}\cdot2$ |
| 24.288.9-24.kj.2.25 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.kl.2.30 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.kr.2.30 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 24.288.9-24.kt.2.31 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{3}\cdot2$ |
| 72.432.16-72.br.1.63 | $72$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 120.288.7-120.bai.1.36 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.baq.1.52 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bay.1.64 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bbg.1.52 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bbo.1.8 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bbw.1.36 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bce.1.50 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.bcm.1.52 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.8-120.ch.1.49 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.cl.1.33 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.cx.1.39 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.db.1.49 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.dn.1.35 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.dr.1.57 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.ed.1.49 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.eh.1.39 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.fz.1.40 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.gd.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.gp.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.gt.1.40 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.hf.1.7 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.hj.1.39 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.hv.1.40 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.hz.1.8 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.il.1.47 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.ip.1.63 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.jb.1.64 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.jf.1.45 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.jr.1.63 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.jv.1.47 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.kh.1.41 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.kl.1.63 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.mf.1.40 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.mj.1.52 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.mv.1.13 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.mz.1.39 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.nl.1.52 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.np.1.40 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.ob.1.40 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.8-120.of.1.60 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.288.9-120.rz.1.39 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.sd.1.55 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.sp.1.52 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.st.1.40 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tf.1.60 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tj.1.40 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tv.1.40 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.tz.1.52 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bcx.1.61 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bdb.1.45 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bdn.1.41 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bdr.1.63 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bed.1.45 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.beh.1.61 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bet.1.61 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.288.9-120.bex.1.33 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.7-168.yw.1.39 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.ze.1.55 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.zm.1.47 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.zu.1.39 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bac.1.59 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bak.1.55 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bas.1.51 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.bba.1.55 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.8-168.bz.1.49 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.cd.1.33 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.cp.1.45 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.ct.1.49 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.df.1.37 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.dj.1.57 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.dv.1.49 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.dz.1.45 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.fr.1.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.fv.1.8 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.gh.1.8 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.gl.1.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.gx.1.7 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.hb.1.39 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.hn.1.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.hr.1.8 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.id.1.31 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.ih.1.63 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.it.1.64 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.ix.1.29 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.jj.1.63 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.jn.1.31 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.jz.1.25 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.kd.1.63 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.lx.1.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.mb.1.52 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.mn.1.13 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.mr.1.39 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.nd.1.52 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.nh.1.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.nt.1.40 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.8-168.nx.1.60 | $168$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 168.288.9-168.rr.1.39 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.rv.1.55 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.sh.1.52 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.288.9-168.sl.1.40 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
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