Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| 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 | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | 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: | 24V3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.3.1697 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&11\\12&7\end{bmatrix}$, $\begin{bmatrix}17&8\\0&19\end{bmatrix}$, $\begin{bmatrix}19&9\\12&1\end{bmatrix}$, $\begin{bmatrix}23&1\\12&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2^2\times D_{12}):C_4$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.ga.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 48.2.a.a, 144.2.a.b$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x u + y u + w t $ |
| $=$ | $x^{2} - x y - x z - y^{2}$ | |
| $=$ | $x t - 5 y t - z t + w u$ | |
| $=$ | $5 x y - 6 y^{2} - y z - w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 25 x^{8} + 904 x^{6} y^{2} - 300 x^{6} z^{2} + 144 x^{4} y^{4} - 726 x^{4} y^{2} z^{2} + 990 x^{4} z^{4} + \cdots + 81 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 9 w^{2} $ | $=$ | $ 9 x^{4} - 3 x^{2} z^{2} + z^{4} $ |
| $0$ | $=$ | $-3 x^{2} + y^{2} + z^{2}$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^6}{11}\cdot\frac{154294564608000yzu^{10}-1876274721345024z^{2}w^{10}+128865022070400z^{2}w^{8}u^{2}+183008786105376z^{2}w^{6}u^{4}-383938625308392z^{2}w^{4}u^{6}+180429989822280z^{2}w^{2}u^{8}-4913551458816z^{2}t^{10}+136214838144z^{2}t^{8}u^{2}+4445584880352z^{2}t^{6}u^{4}+26126840247816z^{2}t^{4}u^{6}+32019692916456z^{2}t^{2}u^{8}+34762245470976z^{2}u^{10}-36082206179712w^{12}-171820029427200w^{10}u^{2}+1218689809317288w^{8}u^{4}-2373289846674516w^{6}u^{6}+930345673512915w^{4}u^{8}-138361886558211w^{2}u^{10}+16445998701440t^{12}-113716891904t^{10}u^{2}+1110206246072t^{8}u^{4}+371950866980t^{6}u^{6}-3025021126911t^{4}u^{8}-2693348172287t^{2}u^{10}+4449120922880u^{12}}{u^{2}(179159040yzu^{8}+312400053504z^{2}w^{8}+141323833728z^{2}w^{6}u^{2}-63135271584z^{2}w^{4}u^{4}+3006441504z^{2}w^{2}u^{6}-2454321408z^{2}t^{8}+3904602240z^{2}t^{6}u^{2}+3633308448z^{2}t^{4}u^{4}-330300768z^{2}t^{2}u^{6}-22394880z^{2}u^{8}+1952500334400w^{10}+1045982322000w^{8}u^{2}+313030963476w^{6}u^{4}+73182262527w^{4}u^{6}+11622847791w^{2}u^{8}+204526784t^{10}-376515216t^{8}u^{2}-23013383316t^{6}u^{4}+8672135885t^{4}u^{6}-3993721957t^{2}u^{8}+7464960u^{10})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.ga.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}u$ |
Equation of the image curve:
| $0$ | $=$ | $ 25X^{8}+904X^{6}Y^{2}+144X^{4}Y^{4}-300X^{6}Z^{2}-726X^{4}Y^{2}Z^{2}-216X^{2}Y^{4}Z^{2}+990X^{4}Z^{4}+396X^{2}Y^{2}Z^{4}+81Y^{4}Z^{4}-540X^{2}Z^{6}-270Y^{2}Z^{6}+81Z^{8} $ |
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.l.1.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.48.0-24.bs.1.4 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.96.1-12.l.1.14 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iu.1.18 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iu.1.29 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iw.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iw.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
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.fr.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.fr.2.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.fr.3.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.fr.4.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.fs.1.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.fs.2.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.fs.3.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.fs.4.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.576.13-24.nw.1.6 | $24$ | $3$ | $3$ | $13$ | $0$ | $1^{10}$ |
| 48.384.9-48.bab.1.2 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{6}$ |
| 48.384.9-48.bad.1.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{6}$ |
| 48.384.9-48.ber.1.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.ber.2.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bes.1.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bes.2.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bgv.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{6}$ |
| 48.384.9-48.bgx.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{6}$ |
| 120.384.5-120.yt.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.yt.2.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.yt.3.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.yt.4.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.yu.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.yu.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.yu.3.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.yu.4.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yt.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yt.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yt.3.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yt.4.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yu.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yu.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yu.3.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yu.4.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.9-240.fkv.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fkw.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmb.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmb.2.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmc.1.25 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmc.2.25 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fof.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fog.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.5-264.yt.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.yt.2.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.yt.3.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.yt.4.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.yu.1.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.yu.2.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.yu.3.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.yu.4.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.yt.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.yt.2.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.yt.3.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.yt.4.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.yu.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.yu.2.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.yu.3.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.yu.4.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |