Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $192$ | ||
| 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: | 24.192.1.3004 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&21\\8&1\end{bmatrix}$, $\begin{bmatrix}11&18\\12&23\end{bmatrix}$, $\begin{bmatrix}17&15\\16&5\end{bmatrix}$, $\begin{bmatrix}23&21\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $S_3\times D_4:D_4$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.dq.3 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: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x z + x w + y z + 2 y w $ |
| $=$ | $2 x^{2} + 8 x y + 2 y^{2} + 3 z^{2} + 3 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + 4 x^{3} z - 2 x^{2} y^{2} + 5 x^{2} z^{2} - 8 x y^{2} z + 4 x z^{3} - 2 y^{2} z^{2} + z^{4} $ |
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{25776703238307840xy^{23}-42432859478163456xy^{21}w^{2}+2322878627189882880xy^{19}w^{4}+197155492968743829504xy^{17}w^{6}+21284337856697692323840xy^{15}w^{8}+2545810038152941000458240xy^{13}w^{10}+324826167102735758250737664xy^{11}w^{12}+43320098846215556267847450624xy^{9}w^{14}+5966305555605536422402353266688xy^{7}w^{16}+842066606218035458521402881343488xy^{5}w^{18}+121153723858408847283815826766823424xy^{3}w^{20}+17703591223404897178992746042427113472xyw^{22}+6906846816239616y^{24}+155253378606170112y^{22}w^{2}+8159867647668781056y^{20}w^{4}+735218158449620680704y^{18}w^{6}+79342297281186254290944y^{16}w^{8}+9491203676254991032516608y^{14}w^{10}+1211085083246015608997806080y^{12}w^{12}+161521918647294883827557597184y^{10}w^{14}+22246432820469313163649381629952y^{8}w^{16}+3139864022876791978217277697425408y^{6}w^{18}+451760723087586397692339811871195136y^{4}w^{20}+66014428358335987780805791600359800832y^{2}w^{22}+24500243672736989183z^{24}-802822739870449925400z^{23}w+14902980213302051535876z^{22}w^{2}-205628907340454742327304z^{21}w^{3}+2333838861822210545604702z^{20}w^{4}-23005415810235715995714600z^{19}w^{5}+203278050869896348673719796z^{18}w^{6}-1645392388020269110494452856z^{17}w^{7}+12374398589820778596766619025z^{16}w^{8}-87440803752614166231008785648z^{15}w^{9}+584662002452989906551771632136z^{14}w^{10}-3724006534999701655982478855504z^{13}w^{11}+22658613137113759743021733231652z^{12}w^{12}-132332929074454799544697676762448z^{11}w^{13}+740533661783843213334595554382344z^{10}w^{14}-3992315049755745065987668769354992z^{9}w^{15}+20506238036984540505250992685094289z^{8}w^{16}-101650756983155634412778029970316408z^{7}w^{17}+466292491003593587276075729374363124z^{6}w^{18}-2082994185171269185314882297937151016z^{5}w^{19}+7460219309054398228749212770666600542z^{4}w^{20}-28631776074234439141263899233071792136z^{3}w^{21}-8342454279369231989702826417683891196z^{2}w^{22}-26646569025737321254713228263751154968zw^{23}-15356168887919811488570451155392401409w^{24}}{wz(z-w)^{6}(z+w)^{2}(z^{2}+w^{2})^{4}(z^{2}+zw+w^{2})^{3}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.1.dq.3 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{4}-2X^{2}Y^{2}+4X^{3}Z-8XY^{2}Z+5X^{2}Z^{2}-2Y^{2}Z^{2}+4XZ^{3}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-12.c.4.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-12.c.4.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bs.2.21 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bs.2.23 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.ix.1.20 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ix.1.22 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.de.2.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.dk.2.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.eq.2.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.er.2.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fm.2.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fo.2.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fu.3.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fw.2.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.9-24.bd.1.3 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.baq.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bas.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bbg.3.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bbi.2.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdc.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bde.2.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bds.3.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdu.2.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.baq.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bas.1.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bbg.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bbi.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdc.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bde.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bds.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdu.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.baq.2.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bas.2.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bbg.4.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bbi.2.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdc.2.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bde.2.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bds.4.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdu.1.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.baq.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bas.1.13 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bbg.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bbi.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdc.2.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bde.1.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bds.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdu.1.13 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |