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: | $6$ (none of which are rational) | Cusp widths | $6^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 6E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.147 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&19\\16&23\end{bmatrix}$, $\begin{bmatrix}7&9\\12&17\end{bmatrix}$, $\begin{bmatrix}11&12\\18&13\end{bmatrix}$, $\begin{bmatrix}21&14\\22&9\end{bmatrix}$ |
| 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: | $2^{6}\cdot3^{2}$ |
| 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 1 rational CM point but no rational cusps or other known rational points.
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\cdot3^3\,\frac{324x^{2}y^{10}+5108832x^{2}y^{8}z^{2}+26753924699136x^{2}y^{6}z^{4}-1923943972914542592x^{2}y^{4}z^{6}+39187467384207693004800x^{2}y^{2}z^{8}-237209586534186717672308736x^{2}z^{10}+23004xy^{10}z+3994126848xy^{8}z^{3}-1404273168052992xy^{6}z^{5}+71290302510353817600xy^{4}z^{7}-1196750853433141507768320xy^{2}z^{9}+6353566197920556894225432576xz^{11}+y^{12}-905904y^{10}z^{2}-418095893376y^{8}z^{4}+51032270127928320y^{6}z^{6}-1491013918978187120640y^{4}z^{8}+14834090040696350805786624y^{2}z^{10}-42084613916122800012997165056z^{12}}{324x^{2}y^{10}-208342368x^{2}y^{8}z^{2}+5036892926976x^{2}y^{6}z^{4}-2556630853632x^{2}y^{4}z^{6}+285637378129920x^{2}y^{2}z^{8}-18509302102818816x^{2}z^{10}-46332xy^{10}z+8756678016xy^{8}z^{3}-135119001795840xy^{6}z^{5}-33853318889472xy^{4}z^{7}+3599030964436992xy^{2}z^{9}-222111625233825792xz^{11}-y^{12}+3775248y^{10}z^{2}-205038283392y^{8}z^{4}+895086910411776y^{6}z^{6}+934668976214016y^{4}z^{8}-108999223494377472y^{2}z^{10}+7329683632716251136z^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 12.18.0.a.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.bc.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.18.0.g.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 |
|---|---|---|---|---|---|---|
| 24.72.3.jk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.jo.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.km.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.kq.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.rh.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.rk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.sj.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.sm.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.7.dj.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.dw.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.3.cht.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.chv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cia.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cic.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cqj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cql.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cqq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cqs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.jo.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.ho.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.cen.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cep.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ceu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cew.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cnd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cnf.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cnk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cnm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.bae.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.3.cen.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cep.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ceu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cew.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cnd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cnf.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cnk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cnm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cen.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cep.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ceu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cew.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cnd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cnf.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cnk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cnm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |