Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $3^{2}\cdot6\cdot24$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-16$) |
Other labels
| Cummins and Pauli (CP) label: | 24B2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.2.49 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&7\\20&1\end{bmatrix}$, $\begin{bmatrix}3&4\\16&21\end{bmatrix}$, $\begin{bmatrix}13&0\\0&1\end{bmatrix}$, $\begin{bmatrix}19&4\\8&13\end{bmatrix}$, $\begin{bmatrix}21&20\\8&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.36.2.cw.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{2}$ |
| Newforms: | 36.2.a.a, 144.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} w + y w^{2} $ |
| $=$ | $x^{2} z + y z w$ | |
| $=$ | $x^{2} y + y^{2} w$ | |
| $=$ | $x^{3} + x y w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + x y^{2} z^{2} - 2 y z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 1 $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:1)$, $(0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{12xyz^{5}w-76xz^{3}w^{4}-16y^{2}w^{6}-48yz^{4}w^{3}-z^{8}+16z^{2}w^{6}}{wz^{5}yx}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.36.2.cw.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}+XY^{2}Z^{2}-2YZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.36.2.cw.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -xzw+\frac{1}{8}w^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.o.1.6 | $8$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.o.1.6 | $8$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 12.36.1-12.c.1.7 | $12$ | $2$ | $2$ | $1$ | $0$ | $1$ |
| 24.36.1-12.c.1.21 | $24$ | $2$ | $2$ | $1$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.3-24.oe.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.og.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.om.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.oo.1.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.3-24.po.1.19 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.pq.1.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.pw.1.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.py.1.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.ci.1.2 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.eu.1.1 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.ew.1.2 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.fo.1.9 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.fq.1.9 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.ga.1.11 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.gc.1.9 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.hg.1.8 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.hi.1.14 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.ho.1.12 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.hq.1.11 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.im.1.7 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.io.1.7 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.iu.1.7 | $24$ | $2$ | $2$ | $4$ | $2$ | $1^{2}$ |
| 24.144.4-24.iw.1.15 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 48.144.4-48.s.1.15 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 48.144.4-48.t.1.13 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 48.144.4-48.w.1.13 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 48.144.4-48.x.1.15 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 48.144.4-48.bi.1.14 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 48.144.4-48.bj.1.16 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 48.144.4-48.bm.1.16 | $48$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 48.144.4-48.bn.1.14 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 48.144.5-48.c.1.19 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 48.144.5-48.d.1.17 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 48.144.5-48.g.1.17 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 48.144.5-48.h.1.19 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 48.144.5-48.k.1.25 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 48.144.5-48.l.1.29 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 48.144.5-48.o.1.29 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 48.144.5-48.p.1.25 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 72.216.8-72.du.1.22 | $72$ | $3$ | $3$ | $8$ | $?$ | not computed |
| 120.144.3-120.ccy.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cda.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cdo.1.14 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cdq.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cfk.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cfm.1.19 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cga.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cgc.1.11 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.4-120.ky.1.38 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.la.1.6 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.lw.1.5 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ly.1.6 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.nk.1.25 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.nm.1.17 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.oi.1.21 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ok.1.17 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.qm.1.13 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.qo.1.28 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.rc.1.21 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.re.1.26 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.sy.1.9 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ta.1.9 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.to.1.9 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.tq.1.27 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.360.14-120.gg.1.56 | $120$ | $5$ | $5$ | $14$ | $?$ | not computed |
| 120.432.15-120.jq.1.110 | $120$ | $6$ | $6$ | $15$ | $?$ | not computed |
| 168.144.3-168.cae.1.28 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cag.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cau.1.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.caw.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.ccq.1.11 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.ccs.1.21 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cdg.1.19 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cdi.1.21 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.4-168.ka.1.22 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.kc.1.6 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ky.1.5 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.la.1.6 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.mm.1.25 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.mo.1.17 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.nk.1.21 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.nm.1.17 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.po.1.21 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.pq.1.28 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.qe.1.13 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.qg.1.22 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.sa.1.9 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.sc.1.9 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.sq.1.9 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ss.1.27 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.4-240.by.1.28 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.4-240.bz.1.20 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.4-240.cc.1.36 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.4-240.cd.1.44 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.4-240.co.1.18 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.4-240.cp.1.26 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.4-240.cs.1.42 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.4-240.ct.1.34 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.144.5-240.c.1.31 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.144.5-240.d.1.23 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.144.5-240.g.1.39 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.144.5-240.h.1.47 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.144.5-240.k.1.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.144.5-240.l.1.25 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.144.5-240.o.1.41 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.144.5-240.p.1.33 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.3-264.cae.1.28 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cag.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cau.1.22 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.caw.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.ccq.1.12 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.ccs.1.22 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cdg.1.20 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cdi.1.22 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.4-264.ka.1.22 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.kc.1.6 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ky.1.5 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.la.1.6 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.mm.1.25 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.mo.1.17 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.nk.1.13 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.nm.1.9 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.po.1.21 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.pq.1.28 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.qe.1.13 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.qg.1.14 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.sa.1.9 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.sc.1.9 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.sq.1.9 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ss.1.27 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.3-312.cae.1.28 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cag.1.23 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cau.1.22 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.caw.1.23 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.ccq.1.23 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.ccs.1.21 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cdg.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cdi.1.13 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.4-312.ky.1.38 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.la.1.6 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.lw.1.5 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ly.1.6 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.nk.1.25 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.nm.1.17 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.oi.1.21 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ok.1.17 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.qm.1.13 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.qo.1.28 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.rc.1.21 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.re.1.26 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.sy.1.9 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ta.1.9 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.to.1.9 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.tq.1.27 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |