Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | 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 | $3^{4}\cdot12^{2}$ | 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: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.115 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&3\\12&11\end{bmatrix}$, $\begin{bmatrix}1&16\\4&11\end{bmatrix}$, $\begin{bmatrix}17&2\\14&1\end{bmatrix}$, $\begin{bmatrix}23&20\\10&5\end{bmatrix}$ |
| 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
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + x y - x z + y^{2} - 2 y z + z^{2} - 2 w^{2} $ |
| $=$ | $x^{2} + 2 x y - 2 x z + 3 y^{2} + 6 y z + 3 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 8 x^{3} z - 6 x^{2} y^{2} + 30 x^{2} z^{2} + 12 x y^{2} z + 8 x z^{3} - 6 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^3\cdot3^3\,\frac{1971216xz^{8}-494424xz^{6}w^{2}+18468xz^{4}w^{4}+1074xz^{2}w^{6}-192xw^{8}-3405888y^{2}z^{7}+1271376y^{2}z^{5}w^{2}-143208y^{2}z^{3}w^{4}+5256y^{2}zw^{6}-15011568yz^{8}+4248288yz^{6}w^{2}-234252yz^{4}w^{4}-14484yz^{2}w^{6}+947yw^{8}-3417552z^{9}+522288z^{7}w^{2}+43092z^{5}w^{4}-3660z^{3}w^{6}-947zw^{8}}{246402xz^{8}-37746xz^{6}w^{2}+1458xz^{4}w^{4}+6xz^{2}w^{6}-425736y^{2}z^{7}+60507y^{2}z^{5}w^{2}-1620y^{2}z^{3}w^{4}-45y^{2}zw^{6}-1876446yz^{8}+332019yz^{6}w^{2}-18225yz^{4}w^{4}+255yz^{2}w^{6}+yw^{8}-427194z^{9}+21546z^{7}w^{2}+4293z^{5}w^{4}-282z^{3}w^{6}-zw^{8}}$ |
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.g.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.d.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.h.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.bv.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.dz.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ib.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ig.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.um.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.us.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.wj.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.wp.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.7.fl.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.ft.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.3.dgx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dgz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dhl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dhn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dxt.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dxv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dyh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.dyj.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.qd.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.ot.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.dct.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dcv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ddh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ddj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dqh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dqj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dqv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dqx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.bgt.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.3.dct.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dcv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ddh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ddj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dqh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dqj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dqv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dqx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dct.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dcv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ddh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ddj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dqh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dqj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dqv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dqx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |