Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $16$ 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: | 24H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.69 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&20\\20&3\end{bmatrix}$, $\begin{bmatrix}5&0\\12&1\end{bmatrix}$, $\begin{bmatrix}13&17\\8&23\end{bmatrix}$, $\begin{bmatrix}19&7\\10&5\end{bmatrix}$, $\begin{bmatrix}23&13\\4&1\end{bmatrix}$ |
| 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: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 36x $ |
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.
| Weierstrass model |
|---|
| $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{12}\cdot3^{12}}\cdot\frac{27216x^{2}y^{20}z^{2}-1088391168x^{2}y^{16}z^{6}+2496682268098560x^{2}y^{12}z^{10}-8076570991169195999232x^{2}y^{8}z^{14}+528311399652622542040989696x^{2}y^{4}z^{18}-1173265820828298418377059205120x^{2}z^{22}+216xy^{22}z-43670016xy^{18}z^{5}-114725136236544xy^{14}z^{9}+90095056280031854592xy^{10}z^{13}-24452208407629817840664576xy^{6}z^{17}+260797365911833610580639350784xy^{2}z^{21}+y^{24}+373248y^{20}z^{4}+1710950916096y^{16}z^{8}-398521269966864384y^{12}z^{12}+388599391937094123257856y^{8}z^{16}-6335094243517335103975981056y^{4}z^{20}+10314424798490535546171949056z^{24}}{z^{8}y^{12}(36xz-y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.0.p.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ce.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gs.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.mq.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.mr.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ms.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.mt.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ni.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.nj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.nk.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.nl.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.pu.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.pv.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.pw.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.px.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.qk.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ql.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.qm.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.qn.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.9.hp.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.oj.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bfq.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bfu.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.blr.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bls.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bmp.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bmq.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 72.216.13.me.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.koc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kod.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.koe.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kof.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kos.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kot.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kou.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kov.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kre.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfuk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bful.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfva.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfvb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfww.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfwx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hob.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hoc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hod.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hoe.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hor.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hos.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hot.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hou.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqo.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hrd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hre.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hrf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hrg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbre.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbrf.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbru.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbrv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbtq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbtr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbug.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbuh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hoc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hod.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hoe.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hof.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hos.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hot.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hou.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hov.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqo.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqp.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hre.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bbxe.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbxf.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbxu.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbxv.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbzq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbzr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcag.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcah.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hoc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hod.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hoe.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hof.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hos.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hot.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hou.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hov.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqo.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hre.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbrm.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbrn.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbsc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbsd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbty.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbtz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbuo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbup.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
