Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.3.1513 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&12\\12&5\end{bmatrix}$, $\begin{bmatrix}9&22\\8&7\end{bmatrix}$, $\begin{bmatrix}17&4\\0&19\end{bmatrix}$, $\begin{bmatrix}21&2\\22&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^5.D_6$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.bh.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{15}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}$ |
| Newforms: | 72.2.a.a, 192.2.a.b, 576.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ 2 x u + z t + w t $ |
| $=$ | $x^{2} - x y + y z - y w + z w$ | |
| $=$ | $x z + x w - 2 y z - 2 y w + z^{2} - w^{2} + t u$ | |
| $=$ | $2 x^{2} + x z - x w - 2 y z + 2 y w + z^{2} + w^{2} + t^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{6} y^{2} + x^{6} z^{2} - 6 x^{5} y^{2} z - 4 x^{5} z^{3} + 9 x^{4} y^{4} + 24 x^{4} y^{2} z^{2} + \cdots + 54 y^{2} z^{6} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 81 w^{2} $ | $=$ | $ -28 x^{4} - 8 x^{3} z - 8 x^{2} y z - 28 x^{2} z^{2} - 56 x y z^{2} - 20 x z^{3} + 12 y z^{3} - 7 z^{4} $ |
| $0$ | $=$ | $x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=23$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3}\cdot\frac{55320192xwu^{10}-33965568yzw^{10}+170201088yzw^{8}u^{2}-36910080yzw^{6}u^{4}-122529024yzw^{4}u^{6}+90125568yzw^{2}u^{8}+32423808yzu^{10}-170201088yw^{9}u^{2}+116038656yw^{7}u^{4}+20694528yw^{5}u^{6}-41416704yw^{3}u^{8}+64568832ywu^{10}+141709824zw^{9}u^{2}-87868800zw^{7}u^{4}+26500608zw^{5}u^{6}-102817728zw^{3}u^{8}+34532352zwu^{10}-34012224w^{12}-5660928w^{10}u^{2}+96391296w^{8}u^{4}-97172352w^{6}u^{6}+48052224w^{4}u^{8}-2124306w^{2}t^{10}+8505972w^{2}t^{9}u-47116728w^{2}t^{8}u^{2}+103817376w^{2}t^{7}u^{3}-164164806w^{2}t^{6}u^{4}+120033900w^{2}t^{5}u^{5}+32131728w^{2}t^{4}u^{6}-188338176w^{2}t^{3}u^{7}+365903424w^{2}t^{2}u^{8}-269910144w^{2}tu^{9}+307009728w^{2}u^{10}-531441t^{12}-1062153t^{10}u^{2}+4961088t^{9}u^{3}-26512677t^{8}u^{4}+60763932t^{7}u^{5}-80884197t^{6}u^{6}+23308992t^{5}u^{7}+111353040t^{4}u^{8}-184052304t^{3}u^{9}+130087344t^{2}u^{10}-24248160tu^{11}-4096u^{12}}{u^{4}(240xwu^{6}+432yzw^{6}-432yzw^{4}u^{2}-576yzw^{2}u^{4}+72yzu^{6}-432yw^{7}-864yw^{5}u^{2}-96yw^{3}u^{4}-64ywu^{6}+252zw^{3}u^{4}+224zwu^{6}+72w^{6}u^{2}-108w^{4}u^{4}-54w^{2}t^{6}-108w^{2}t^{5}u-90w^{2}t^{4}u^{2}-144w^{2}t^{3}u^{3}+6w^{2}t^{2}u^{4}+240w^{2}tu^{5}-244w^{2}u^{6}-27t^{6}u^{2}-36t^{5}u^{3}+24t^{4}u^{4}-18t^{3}u^{5}-16t^{2}u^{6}-2tu^{7})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.bh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{6}Y^{2}+9X^{4}Y^{4}-6X^{5}Y^{2}Z+X^{6}Z^{2}+24X^{4}Y^{2}Z^{2}+54X^{2}Y^{4}Z^{2}-4X^{5}Z^{3}-36X^{3}Y^{2}Z^{3}+10X^{4}Z^{4}+72X^{2}Y^{2}Z^{4}+81Y^{4}Z^{4}-12X^{3}Z^{5}-54XY^{2}Z^{5}+9X^{2}Z^{6}+54Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.1-12.d.1.6 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.48.0-24.g.1.5 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.96.1-12.d.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.bx.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.bx.1.8 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.by.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.by.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.5-24.bx.1.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.bx.2.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.bx.3.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.bx.4.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ca.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ca.2.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ca.3.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ca.4.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.576.13-24.cz.1.1 | $24$ | $3$ | $3$ | $13$ | $2$ | $1^{10}$ |
| 120.384.5-120.ij.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ij.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ij.3.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ij.4.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.im.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.im.2.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.im.3.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.im.4.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ij.1.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ij.2.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ij.3.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ij.4.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.im.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.im.2.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.im.3.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.im.4.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ij.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ij.2.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ij.3.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ij.4.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.im.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.im.2.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.im.3.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.im.4.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ij.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ij.2.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ij.3.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ij.4.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.im.1.12 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.im.2.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.im.3.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.im.4.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |