Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ 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: | 12D3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.144.3.26 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&6\\0&7\end{bmatrix}$, $\begin{bmatrix}1&15\\18&13\end{bmatrix}$, $\begin{bmatrix}11&3\\12&5\end{bmatrix}$, $\begin{bmatrix}17&9\\12&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.3.ug.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $512$ |
Jacobian
| Conductor: | $2^{18}\cdot3^{4}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 192.2.a.d$^{2}$, 576.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - x z t + x w t - y^{2} t + y w t - z w t $ |
| $=$ | $x^{2} y - x y z + x y w - y^{3} + y^{2} w - y z w$ | |
| $=$ | $x z t - x w t + 2 y z t + y w t + z^{2} t - w^{2} t$ | |
| $=$ | $x^{3} - x^{2} z - x y^{2} + x y w - x w^{2} + y^{2} w - y w^{2} + z w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 6 x^{5} z + 12 x^{4} y^{2} + 30 x^{4} z^{2} + 48 x^{3} y^{2} z + 80 x^{3} z^{3} + 36 x^{2} y^{4} + \cdots + 52 z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -2x^{7} - 14x^{4} + 16x $ |
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.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(-1/2:1:-1/2:1:0)$, $(-1:1:0:0:0)$, $(-1/2:0:-1/2:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2\,\frac{160512xw^{10}+393216xw^{8}t^{2}+260736xw^{6}t^{4}-17712xw^{4}t^{6}-92730xw^{2}t^{8}-19821xt^{10}-192y^{7}t^{4}-1248y^{5}t^{6}-3840y^{3}t^{8}-236544yzw^{9}-505344yzw^{7}t^{2}-493056yzw^{5}t^{4}-250056yzw^{3}t^{6}-40380yzwt^{8}-133632yw^{10}-334848yw^{8}t^{2}-481728yw^{6}t^{4}-393084yw^{4}t^{6}-145524yw^{2}t^{8}-25029yt^{10}-233472z^{2}w^{9}-569856z^{2}w^{7}t^{2}-543744z^{2}w^{5}t^{4}-243872z^{2}w^{3}t^{6}-28944z^{2}wt^{8}+65280zw^{10}+118272zw^{8}t^{2}+216960zw^{6}t^{4}+198952zw^{4}t^{6}+76782zw^{2}t^{8}+7305zt^{10}+185088w^{11}+437376w^{9}t^{2}+496320w^{7}t^{4}+305452w^{5}t^{6}+76554w^{3}t^{8}+12300wt^{10}}{t^{6}(84xw^{4}-72xw^{2}t^{2}+21xt^{4}-48yzw^{3}+24yzwt^{2}+36yw^{4}-36yw^{2}t^{2}+21yt^{4}-48z^{2}w^{3}+32z^{2}wt^{2}+20zw^{2}t^{2}-12zt^{4}+2w^{3}t^{2}-9wt^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.72.3.ug.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{4}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}+12X^{4}Y^{2}+36X^{2}Y^{4}+6X^{5}Z+48X^{3}Y^{2}Z+72XY^{4}Z+30X^{4}Z^{2}+108X^{2}Y^{2}Z^{2}-288Y^{4}Z^{2}+80X^{3}Z^{3}+120XY^{2}Z^{3}+141X^{2}Z^{4}+48Y^{2}Z^{4}+138XZ^{5}+52Z^{6} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.72.3.ug.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -\frac{2}{3}z^{5}-\frac{2}{3}z^{4}w+\frac{5}{6}z^{3}w^{2}-\frac{1}{4}z^{3}t^{2}+\frac{13}{6}z^{2}w^{3}-\frac{5}{6}zw^{4}+\frac{3}{4}zw^{2}t^{2}-\frac{5}{6}w^{5}-\frac{1}{2}w^{3}t^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 6z^{13}w^{6}t+9z^{12}w^{7}t-54z^{11}w^{8}t+\frac{9}{4}z^{11}w^{6}t^{3}-\frac{21}{2}z^{10}w^{9}t+\frac{9}{8}z^{10}w^{7}t^{3}+\frac{819}{8}z^{9}w^{10}t-\frac{351}{16}z^{9}w^{8}t^{3}+\frac{1053}{16}z^{8}w^{11}t+\frac{351}{32}z^{8}w^{9}t^{3}-\frac{2745}{16}z^{7}w^{12}t+\frac{1917}{32}z^{7}w^{10}t^{3}-\frac{999}{16}z^{6}w^{13}t-\frac{2187}{32}z^{6}w^{11}t^{3}+\frac{2187}{16}z^{5}w^{14}t-\frac{675}{32}z^{5}w^{12}t^{3}+\frac{507}{16}z^{4}w^{15}t+\frac{1647}{32}z^{4}w^{13}t^{3}-\frac{819}{16}z^{3}w^{16}t-\frac{81}{32}z^{3}w^{14}t^{3}-\frac{189}{16}z^{2}w^{17}t-\frac{477}{32}z^{2}w^{15}t^{3}+\frac{123}{16}zw^{18}t+\frac{45}{32}zw^{16}t^{3}+\frac{9}{4}w^{19}t+\frac{27}{16}w^{17}t^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z^{3}w^{2}-\frac{3}{2}z^{2}w^{3}+\frac{1}{2}w^{5}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.72.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.et.1.6 | $24$ | $3$ | $3$ | $1$ | $0$ | $1^{2}$ |
| 24.48.1-24.et.1.8 | $24$ | $3$ | $3$ | $1$ | $0$ | $1^{2}$ |
| 24.72.0-6.a.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.288.5-24.a.1.6 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.288.5-24.y.1.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.288.5-24.dn.1.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.288.5-24.dq.1.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.288.5-24.iu.1.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 24.288.5-24.iv.1.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.288.5-24.jb.1.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.288.5-24.jd.1.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}$ |
| 72.432.11-72.ez.1.6 | $72$ | $3$ | $3$ | $11$ | $?$ | not computed |
| 72.432.11-72.ez.1.10 | $72$ | $3$ | $3$ | $11$ | $?$ | not computed |
| 72.432.11-72.ge.1.4 | $72$ | $3$ | $3$ | $11$ | $?$ | not computed |
| 72.432.11-72.hq.1.4 | $72$ | $3$ | $3$ | $11$ | $?$ | not computed |
| 72.432.13-72.en.1.4 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 72.432.13-72.en.1.6 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 72.432.15-72.wo.1.4 | $72$ | $3$ | $3$ | $15$ | $?$ | not computed |
| 120.288.5-120.czk.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.czl.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.czr.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.czs.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dro.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.drp.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.drv.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.drw.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bpk.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bpl.1.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bpr.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bps.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.cae.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.caf.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.cal.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.cam.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.bpk.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.bpl.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.bpr.1.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.bps.1.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.cae.1.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.caf.1.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.cal.1.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.cam.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bpk.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bpl.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bpr.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bps.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.cae.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.caf.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.cal.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.cam.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |