Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $64$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.364 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&3\\18&23\end{bmatrix}$, $\begin{bmatrix}7&4\\16&11\end{bmatrix}$, $\begin{bmatrix}9&14\\10&15\end{bmatrix}$, $\begin{bmatrix}17&8\\14&7\end{bmatrix}$, $\begin{bmatrix}19&13\\2&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} + z w $ |
| $=$ | $6 y^{2} + 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 6 y^{2} z^{2} - 9 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(4z^{3}-w^{3})^{3}(4z^{3}+w^{3})^{3}}{w^{6}z^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.36.0.cb.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ce.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gp.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.5.rq.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.rr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.rs.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.rt.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ti.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.tj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tk.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tl.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vi.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.vj.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vk.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.vl.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vz.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.wa.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.wb.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.fh.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.oe.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bhi.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bhq.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bty.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bua.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.buo.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.buq.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 72.216.13.mu.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kta.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kts.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwe.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfzi.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgck.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hsz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hta.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hts.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvo.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwe.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbwc.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbws.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwt.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbze.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzf.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hta.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hts.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvo.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvp.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwe.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bccc.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccs.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcct.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bceo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcep.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfe.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcff.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hta.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hts.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvo.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwe.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbwk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxa.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzm.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzn.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |