Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $8$ 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: | 12T1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.131 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&9\\6&11\end{bmatrix}$, $\begin{bmatrix}3&10\\22&21\end{bmatrix}$, $\begin{bmatrix}7&12\\6&17\end{bmatrix}$, $\begin{bmatrix}7&21\\18&17\end{bmatrix}$, $\begin{bmatrix}13&12\\18&19\end{bmatrix}$, $\begin{bmatrix}21&22\\22&3\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{4}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x y + 6 y^{2} - w^{2} $ |
| $=$ | $6 x^{2} + 12 x y - 12 y^{2} + z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} - 4 x^{2} z^{2} - 12 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(z^{3}-4w^{3})^{3}(z^{3}+4w^{3})^{3}}{w^{12}z^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.1.bq.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.36.0.c.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.e.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
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.h.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bv.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.dp.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ds.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.iy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.iz.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.jk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.jm.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.jw.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.jx.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ke.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.kf.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ki.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.kj.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.kk.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.kl.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.km.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.kn.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ko.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.kp.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ku.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.kv.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.lc.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ld.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.te.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tf.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.bew.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bey.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.cmi.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.cmk.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dyo.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dyp.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 72.216.9.bg.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.bs.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.cq.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.144.5.ewy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ewz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.exf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.exg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ezc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ezd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ezj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ezk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.fku.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.fkv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.fky.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.fkz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.flc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.fld.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.fle.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.flf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.flg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.flh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.fli.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.flj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.flo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.flp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.fls.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.flt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.htc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.htd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ioa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.iob.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bcju.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bcjv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bdrq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bdrr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.ciy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ciz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cjf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cjg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.clc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cld.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.clj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.clk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqs.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cqx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.crc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.crd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.crg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.crh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.etc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.etd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fnz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.foa.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.ygo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.ygp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.zok.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.zol.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.ciy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ciz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cjf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cjg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.clc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cld.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.clj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.clk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqs.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cqx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.crc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.crd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.crg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.crh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.etc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.etd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.foa.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fob.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.ymo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.ymp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.zuk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.zul.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.ciy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ciz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cjf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cjg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.clc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cld.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.clj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.clk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqs.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cqx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.crc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.crd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.crg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.crh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.etc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.etd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.foa.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fob.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.ygw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.ygx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.zos.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.zot.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |