Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $6^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ 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: | 6E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.2 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&2\\4&3\end{bmatrix}$, $\begin{bmatrix}3&7\\22&15\end{bmatrix}$, $\begin{bmatrix}15&7\\20&21\end{bmatrix}$, $\begin{bmatrix}17&21\\6&5\end{bmatrix}$, $\begin{bmatrix}21&14\\8&3\end{bmatrix}$ |
| 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: | $2048$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 540x + 4752 $ |
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 |
|---|
| $(12:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3\cdot3^3}\cdot\frac{108x^{2}y^{10}+489888x^{2}y^{8}z^{2}-743552566272x^{2}y^{6}z^{4}-28888253612034048x^{2}y^{4}z^{6}+8735724105314844672x^{2}y^{2}z^{8}+8785540224183143074037760x^{2}z^{10}-7452xy^{10}z-438379776xy^{8}z^{3}+11141180142336xy^{6}z^{5}+947530768920846336xy^{4}z^{7}+5215260881828630347776xy^{2}z^{9}-235317266803535524085366784xz^{11}-y^{12}+206064y^{10}z^{2}+25206837120y^{8}z^{4}+544164564851712y^{6}z^{6}-11058356410798583808y^{4}z^{8}-290362365817326153695232y^{2}z^{10}+1558689411359058313467396096z^{12}}{z^{2}(108x^{2}y^{8}-14941584x^{2}y^{6}z^{2}+487599243264x^{2}y^{4}z^{4}-5844503843831808x^{2}y^{2}z^{6}+23538077210574323712x^{2}z^{8}-7236xy^{8}z+614972736xy^{6}z^{3}-16233958932480xy^{4}z^{5}+171142319079161856xy^{2}z^{7}-630458211648619413504xz^{9}-y^{10}+339984y^{8}z^{2}-15844984128y^{6}z^{4}+265579054497792y^{4}z^{6}-1818991484649603072y^{2}z^{8}+4176015421460730347520z^{10})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.18.0.b.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.bo.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.18.0.l.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.e.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 |
|---|---|---|---|---|---|---|
| 24.72.1.bn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.bo.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.br.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.bs.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.3.lo.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ls.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.mq.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.mu.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.tf.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.tg.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.tj.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.tk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.uv.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.uy.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.vx.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.wa.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.4.k.1 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 72.108.7.ex.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 120.72.1.es.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.et.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.eu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.ev.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.ddx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ddz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dee.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.deg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dou.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dov.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dow.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dox.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dut.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.duv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dva.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dvc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.pk.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.oa.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.da.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.db.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.dc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.dd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.3.czt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.czv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.daa.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dac.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.djq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.djr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.djs.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.djt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dnh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dnj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dno.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dnq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.bga.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.1.cw.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.cx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.cy.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.cz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.3.czt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.czv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.daa.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dac.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.djq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.djr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.djs.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.djt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dnh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dnj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dno.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dnq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.1.da.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.db.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.dc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.dd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.3.czt.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.czv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.daa.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dac.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.djq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.djr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.djs.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.djt.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dnh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dnj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dno.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dnq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |