Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $36$ | ||
| 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: | none |
Other labels
| Cummins and Pauli (CP) label: | 24B2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.2.1019 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&23\\20&11\end{bmatrix}$, $\begin{bmatrix}9&2\\8&3\end{bmatrix}$, $\begin{bmatrix}17&8\\8&1\end{bmatrix}$, $\begin{bmatrix}19&1\\16&17\end{bmatrix}$, $\begin{bmatrix}19&22\\16&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.36.2.cl.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: | $2^{4}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{2}$ |
| Newforms: | 36.2.a.a$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} w - z w^{2} $ |
| $=$ | $3 x^{2} z - z^{2} w$ | |
| $=$ | $3 x^{2} y - y z w$ | |
| $=$ | $3 x^{3} - x z w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + 27 x y^{2} z^{2} + 2 y z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 27 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:1)$, $(0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^5\,\frac{1440xz^{6}w-8284xz^{3}w^{4}+512xw^{7}+72y^{5}w^{3}-128yz^{7}+5416yz^{4}w^{3}-2025yzw^{6}}{wz(64xz^{5}-4xz^{2}w^{3}-32yz^{3}w^{2}+yw^{5})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.36.2.cl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{2}{9}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}+27XY^{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.cl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{1}{2}w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -3xyw+\frac{1}{8}w^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.24.0-24.z.1.6 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.36.1-12.c.1.17 | $24$ | $2$ | $2$ | $1$ | $0$ | $1$ |
| 24.36.1-12.c.1.24 | $24$ | $2$ | $2$ | $1$ | $0$ | $1$ |
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.nt.1.4 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.nv.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.nx.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.nz.1.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.oz.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.pb.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.pf.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1$ |
| 24.144.3-24.ph.1.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1$ |
| 24.144.4-24.cd.1.16 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.co.1.11 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.dl.1.9 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.dm.1.9 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.fd.1.18 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.fe.1.9 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.fo.1.12 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.fr.1.10 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.gv.1.4 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.gx.1.1 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.gz.1.12 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.hb.1.1 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.ib.1.1 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.id.1.2 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 24.144.4-24.if.1.9 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{2}$ |
| 24.144.4-24.ih.1.2 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}$ |
| 72.216.8-72.dj.1.7 | $72$ | $3$ | $3$ | $8$ | $?$ | not computed |
| 120.144.3-120.cbp.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cbr.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cbt.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cbv.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ceb.1.16 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ced.1.12 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.cef.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.3-120.ceh.1.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.144.4-120.jx.1.21 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.jz.1.26 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.kb.1.21 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.kd.1.18 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.mj.1.26 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ml.1.21 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.mn.1.26 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.mp.1.22 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.pd.1.16 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.pf.1.6 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.ph.1.32 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.pj.1.10 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.rp.1.6 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.rr.1.8 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.rt.1.30 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.144.4-120.rv.1.14 | $120$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 120.360.14-120.ff.1.47 | $120$ | $5$ | $5$ | $14$ | $?$ | not computed |
| 120.432.15-120.hz.1.85 | $120$ | $6$ | $6$ | $15$ | $?$ | not computed |
| 168.144.3-168.byv.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.byx.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.byz.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.bzb.1.28 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cbh.1.24 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cbj.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cbl.1.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.3-168.cbn.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.4-168.iz.1.25 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.jb.1.26 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.jd.1.21 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.jf.1.18 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ll.1.26 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ln.1.25 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.lp.1.26 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.lr.1.22 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.of.1.16 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.oh.1.4 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.oj.1.32 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.ol.1.6 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.qr.1.4 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.qt.1.8 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.qv.1.28 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 168.144.4-168.qx.1.14 | $168$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.3-264.byv.1.8 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.byx.1.31 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.byz.1.4 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.bzb.1.27 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cbh.1.23 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cbj.1.13 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cbl.1.21 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.3-264.cbn.1.25 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.144.4-264.iz.1.21 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.jb.1.26 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.jd.1.21 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.jf.1.18 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ll.1.26 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ln.1.21 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.lp.1.26 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.lr.1.14 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.of.1.16 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.oh.1.4 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.oj.1.32 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.ol.1.6 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.qr.1.6 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.qt.1.8 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.qv.1.30 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 264.144.4-264.qx.1.14 | $264$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.3-312.byv.1.12 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.byx.1.32 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.byz.1.12 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.bzb.1.28 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cbh.1.16 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cbj.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cbl.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.3-312.cbn.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.144.4-312.jx.1.21 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.jz.1.26 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.kb.1.21 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.kd.1.18 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.mj.1.26 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ml.1.25 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.mn.1.26 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.mp.1.22 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.pd.1.16 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.pf.1.6 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.ph.1.32 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.pj.1.10 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.rp.1.4 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.rr.1.8 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.rt.1.28 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 312.144.4-312.rv.1.14 | $312$ | $2$ | $2$ | $4$ | $?$ | not computed |