Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $6^{4}\cdot24^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.144.4.2396 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&13\\20&3\end{bmatrix}$, $\begin{bmatrix}5&21\\12&11\end{bmatrix}$, $\begin{bmatrix}7&5\\16&13\end{bmatrix}$, $\begin{bmatrix}15&19\\16&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.es.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: | $2^{18}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 36.2.a.a, 144.2.a.a, 576.2.a.e$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 47 y^{2} - 2 y w + 2 z^{2} - w^{2} $ |
| $=$ | $6 x^{3} + y^{2} z + y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{6} - 9 y^{4} z^{2} + 2 y^{2} z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4\cdot47^2}\cdot\frac{12740283936775152yz^{10}w+194457865180391856yz^{8}w^{3}+1460547749507435136yz^{6}w^{5}+232352240255367552yz^{4}w^{7}+11950397989436160yz^{2}w^{9}+201317063351040yw^{11}+209316995573711z^{12}-4668815579171688z^{10}w^{2}-46158085138335804z^{8}w^{4}+255282450400317888z^{6}w^{6}+35494324739342928z^{4}w^{8}+1687695532520832z^{2}w^{10}+27262714491072w^{12}}{z^{2}(4059894592yz^{8}w-17086774048yz^{6}w^{3}+20423707120yz^{4}w^{5}-9660535288yz^{2}w^{7}+1606597872yw^{9}-78074896z^{10}+2687769824z^{8}w^{2}-5717369144z^{6}w^{4}+4660342904z^{4}w^{6}-1691956217z^{2}w^{8}+230017752w^{10})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.es.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}y+\frac{1}{6}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{6}-9Y^{4}Z^{2}+2Y^{2}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.0-24.be.1.8 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.72.2-24.bu.1.8 | $24$ | $2$ | $2$ | $2$ | $1$ | $1^{2}$ |
| 24.72.2-24.bu.1.11 | $24$ | $2$ | $2$ | $2$ | $1$ | $1^{2}$ |
| 24.72.2-24.cu.1.16 | $24$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 24.72.2-24.cu.1.27 | $24$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
| 24.72.2-24.cv.1.14 | $24$ | $2$ | $2$ | $2$ | $1$ | $1^{2}$ |
| 24.72.2-24.cv.1.21 | $24$ | $2$ | $2$ | $2$ | $1$ | $1^{2}$ |
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.rc.1.6 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 24.288.7-24.rd.1.4 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 24.288.7-24.rk.1.4 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 24.288.7-24.rl.1.6 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 24.288.7-24.to.1.4 | $24$ | $2$ | $2$ | $7$ | $3$ | $1^{3}$ |
| 24.288.7-24.tp.1.6 | $24$ | $2$ | $2$ | $7$ | $3$ | $1^{3}$ |
| 24.288.7-24.tw.1.8 | $24$ | $2$ | $2$ | $7$ | $3$ | $1^{3}$ |
| 24.288.7-24.tx.1.6 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
| 72.432.16-72.es.1.12 | $72$ | $3$ | $3$ | $16$ | $?$ | not computed |
| 120.288.7-120.cwl.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.cwm.1.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.cwt.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.cwu.1.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dbj.1.6 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dbk.1.16 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dbr.1.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.288.7-120.dbs.1.12 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cmu.1.16 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cmv.1.6 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cnc.1.12 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cnd.1.12 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cqm.1.6 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cqn.1.16 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cqu.1.16 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.288.7-168.cqv.1.12 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cmu.1.16 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cmv.1.6 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cnc.1.12 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cnd.1.12 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cqm.1.6 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cqn.1.16 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cqu.1.16 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.288.7-264.cqv.1.12 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.cqu.1.16 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.cqv.1.4 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.crc.1.16 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.crd.1.12 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.cvs.1.4 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.cvt.1.16 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.cwa.1.12 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.cwb.1.12 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |