Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 12D2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.2.81 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&15\\12&5\end{bmatrix}$, $\begin{bmatrix}5&8\\10&17\end{bmatrix}$, $\begin{bmatrix}7&13\\4&11\end{bmatrix}$, $\begin{bmatrix}21&14\\2&3\end{bmatrix}$ |
| 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: | $2^{12}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{2}$ |
| Newforms: | 576.2.a.b, 576.2.a.e |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x w - x t + 2 y w + z t $ |
| $=$ | $2 x w - 2 y w - 2 y t - z t$ | |
| $=$ | $2 x^{2} - 4 x y - 3 x z - 4 y^{2}$ | |
| $=$ | $6 x^{2} + 6 x z + 6 z^{2} - 3 w^{2} + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 27 x^{6} - 6 x^{4} y^{2} - 27 x^{4} z^{2} + 6 x^{2} y^{2} z^{2} + 9 x^{2} z^{4} - 2 y^{2} z^{4} - z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -2x^{6} + 12x^{4} - 24x^{2} + 18 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ | $=$ | $\displaystyle w^{2}t-\frac{1}{3}t^{3}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 6zw^{8}-10zw^{6}t^{2}+\frac{20}{3}zw^{4}t^{4}-2zw^{2}t^{6}+\frac{2}{9}zt^{8}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -w^{3}+\frac{1}{3}wt^{2}$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3\,\frac{108xz^{3}t^{2}-54xzt^{4}-162z^{6}+216z^{4}t^{2}-72z^{2}t^{4}+21w^{6}-54w^{4}t^{2}+45w^{2}t^{4}-8t^{6}}{12xz^{3}t^{2}+2xzt^{4}-72z^{6}+24z^{4}t^{2}+9w^{6}-6w^{4}t^{2}+w^{2}t^{4}}$ |
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.h.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.c.1 | $24$ | $2$ | $2$ | $1$ | $1$ | $1$ |
| 24.18.1.j.1 | $24$ | $2$ | $2$ | $1$ | $1$ | $1$ |
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.g.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 24.72.3.em.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 24.72.3.ep.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 24.72.3.eu.1 | $24$ | $2$ | $2$ | $3$ | $3$ | $1$ |
| 24.72.3.lk.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 24.72.3.lm.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 24.72.3.ly.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1$ |
| 24.72.3.ma.1 | $24$ | $2$ | $2$ | $3$ | $3$ | $1$ |
| 72.108.8.cy.1 | $72$ | $3$ | $3$ | $8$ | $?$ | not computed |
| 72.324.22.fe.1 | $72$ | $9$ | $9$ | $22$ | $?$ | not computed |
| 120.72.3.bua.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.buc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.buo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.buq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bwe.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bwg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bws.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bwu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.14.hu.1 | $120$ | $5$ | $5$ | $14$ | $?$ | not computed |
| 120.216.15.le.1 | $120$ | $6$ | $6$ | $15$ | $?$ | not computed |
| 168.72.3.brs.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bru.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bsg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bsi.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.btw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bty.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.buk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bum.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.21.hu.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.3.brs.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bru.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bsg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bsi.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.btw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bty.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.buk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bum.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.brs.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bru.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bsg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bsi.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.btw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bty.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.buk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bum.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |