Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $12\cdot24$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $8$ 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: | 24D1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.69 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&16\\20&21\end{bmatrix}$, $\begin{bmatrix}11&13\\22&7\end{bmatrix}$, $\begin{bmatrix}13&22\\4&11\end{bmatrix}$, $\begin{bmatrix}15&19\\14&3\end{bmatrix}$, $\begin{bmatrix}19&16\\20&13\end{bmatrix}$ |
| 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: | $2048$ |
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 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^{12}}\cdot\frac{19440x^{2}y^{8}z^{2}-997691904x^{2}y^{4}z^{6}+1234235584512x^{2}z^{10}-216xy^{10}z+53747712xy^{6}z^{5}-362434258944xy^{2}z^{9}+y^{12}-898128y^{8}z^{4}+9115276032y^{4}z^{8}+1586874322944z^{12}}{z^{8}(36xz-y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.0.k.1 | $12$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.72.1.cu.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.cw.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.cy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.da.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.er.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.es.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.ev.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.ew.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.p.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.u.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.bn.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.bq.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.72.5.dd.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.de.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.dh.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.di.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.mq.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.ms.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.mu.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.72.5.mw.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.72.5.ng.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.ni.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.nk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.nm.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.108.7.gj.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.gq.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.1.ww.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.wy.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.xa.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.xc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.yc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ye.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.yg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.yi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.cbs.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cbu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cbw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cby.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cci.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cck.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ccm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cco.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cdo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cdq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cds.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cdu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cee.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ceg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cei.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cek.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.180.13.bwm.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.che.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.ky.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.la.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.lc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.le.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.me.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.mg.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.mi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.mk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.5.bmm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bmo.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bmq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bms.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bnc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bne.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bng.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bni.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.boi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bok.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bom.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.boo.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.boy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bpa.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bpc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.bpe.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.23.bs.1 | $168$ | $8$ | $8$ | $23$ | $?$ | not computed |
| 264.72.1.ku.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.kw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.ky.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.la.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.ma.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.mc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.me.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.mg.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.5.bmm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bmo.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bmq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bms.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bnc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bne.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bng.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bni.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.boi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bok.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bom.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.boo.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.boy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bpa.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bpc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.bpe.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.1.ky.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.la.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.lc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.le.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.me.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.mg.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.mi.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.mk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.5.bmm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bmo.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bmq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bms.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bnc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bne.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bng.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bni.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.boi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bok.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bom.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.boo.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.boy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bpa.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bpc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.bpe.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |