Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $2^{6}\cdot4$ | ||
| 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: | 24J1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.1735 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&3\\8&19\end{bmatrix}$, $\begin{bmatrix}13&3\\16&5\end{bmatrix}$, $\begin{bmatrix}13&21\\0&19\end{bmatrix}$, $\begin{bmatrix}19&18\\12&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2\times D_8):D_6$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.dm.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x z - y w $ |
| $=$ | $6 x^{2} + 6 y^{2} + z^{2} + 4 z w + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} z + 6 x^{2} y^{2} + x^{2} z^{2} + 6 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{35664401793024xy^{21}w^{2}+65384736620544xy^{19}w^{4}+9576552333312xy^{17}w^{6}-34838836936704xy^{15}w^{8}-14915205070848xy^{13}w^{10}+18715861057536xy^{11}w^{12}+18680698109952xy^{9}w^{14}-26240477036544xy^{7}w^{16}-56341297889280xy^{5}w^{18}+18639793815552xy^{3}w^{20}+148126999707648xyw^{22}-8916100448256y^{24}-17832200896512y^{22}w^{2}+25262284603392y^{20}w^{4}+50359456235520y^{18}w^{6}+35506168528896y^{16}w^{8}+31307253940224y^{14}w^{10}+13699025731584y^{12}w^{12}-27961932644352y^{10}w^{14}-38453820456960y^{8}w^{16}+32283581349888y^{6}w^{18}+108592112959488y^{4}w^{20}+4595457687552y^{2}w^{22}+4095z^{24}+196584z^{23}w+4275972z^{22}w^{2}+55409912z^{21}w^{3}+471058014z^{20}w^{4}+2701028568z^{19}w^{5}+10068276468z^{18}w^{6}+19529032840z^{17}w^{7}-15352279663z^{16}w^{8}-214389838576z^{15}w^{9}-546646697976z^{14}w^{10}-178260106064z^{13}w^{11}+2231407300644z^{12}w^{12}+4599350338736z^{11}w^{13}-1540849838072z^{10}w^{14}-16814929269488z^{9}w^{15}-12403098301039z^{8}w^{16}+33057414505608z^{7}w^{17}+51649737635060z^{6}w^{18}-35740411138856z^{5}w^{19}-105909957114274z^{4}w^{20}+2545866407160z^{3}w^{21}+99517242752772z^{2}w^{22}+24687833284584zw^{23}-w^{24}}{w^{3}z^{3}(z-w)^{2}(z+w)^{6}(z^{2}+zw+w^{2})(z^{2}+4zw+w^{2})^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.dm.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}+6X^{2}Y^{2}+4X^{3}Z+X^{2}Z^{2}+6Y^{2}Z^{2} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.c.3.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-12.c.3.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bs.2.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bs.2.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.iv.1.17 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.iv.1.29 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.384.5-24.cw.4.8 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.dj.3.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.ek.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.el.3.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.eu.1.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.ey.3.2 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fu.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fv.3.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.9-24.u.2.1 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.bai.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bak.3.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bay.3.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bba.3.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bcu.1.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bcw.3.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdk.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdm.3.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bai.3.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bak.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bay.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bba.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bcu.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bcw.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdk.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdm.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bai.4.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bak.3.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bay.1.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bba.3.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bcu.1.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bcw.3.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdk.1.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdm.4.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bai.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bak.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bay.2.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bba.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bcu.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bcw.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdk.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdm.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |