Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.9 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&21\\6&5\end{bmatrix}$, $\begin{bmatrix}11&12\\6&1\end{bmatrix}$, $\begin{bmatrix}13&12\\0&5\end{bmatrix}$, $\begin{bmatrix}17&21\\0&19\end{bmatrix}$, $\begin{bmatrix}21&23\\14&21\end{bmatrix}$ |
| 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: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 17x + 15 $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(0:1:0)$, $(3:0:1)$, $(1:0:1)$, $(-5:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6}\cdot\frac{12x^{2}y^{10}+352x^{2}y^{8}z^{2}-187392x^{2}y^{6}z^{4}-21682176x^{2}y^{4}z^{6}+8505769984x^{2}y^{2}z^{8}-439804166144x^{2}z^{10}+36xy^{10}z-64xy^{8}z^{3}+828672xy^{6}z^{5}+227487744xy^{4}z^{7}-42616963072xy^{2}z^{9}+1762333425664xz^{11}+y^{12}+16y^{10}z^{2}+48480y^{8}z^{4}+1068800y^{6}z^{6}-1409361920y^{4}z^{8}+106381672448y^{2}z^{10}-1327689039872z^{12}}{z^{4}y^{6}(4x^{2}-20xz-y^{2}+16z^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 8.6.0.f.1 | $8$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
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 |
| 24.18.0.p.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.k.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.co.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.cp.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.cr.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.cs.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.3.z.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.cc.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ht.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.hw.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ug.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.uh.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.uj.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.uk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.vw.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.vx.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.wd.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.we.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.5.d.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.108.5.cc.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.72.1.qm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.qn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.qp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.qq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.ewi.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ewk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eww.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ewy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eyo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eyp.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eyr.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eys.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ezw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ezy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fak.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.fam.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.bsc.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bxw.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.ig.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.ih.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.ij.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.ik.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.3.ekq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eks.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ele.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.elg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.emw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.emx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.emz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ena.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eoe.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eog.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eos.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eou.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.21.bcw.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.1.ic.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.id.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.if.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.ig.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.3.ekq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eks.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ele.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.elg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.emw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.emx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.emz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ena.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eoe.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eog.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eos.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eou.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.1.ig.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.ih.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.ij.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.ik.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.3.ekq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eks.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ele.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.elg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.emw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.emx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.emz.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ena.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eoe.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eog.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eos.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eou.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |