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Properties

Label 24.96.1.dq.3
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $192$
Index: $96$ $\PSL_2$-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: 24.96.1.1639

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&3\\12&23\end{bmatrix}$, $\begin{bmatrix}1&18\\12&13\end{bmatrix}$, $\begin{bmatrix}11&0\\8&19\end{bmatrix}$, $\begin{bmatrix}19&3\\12&7\end{bmatrix}$, $\begin{bmatrix}19&9\\4&23\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035859
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.dq.3.1, 24.192.1-24.dq.3.2, 24.192.1-24.dq.3.3, 24.192.1-24.dq.3.4, 24.192.1-24.dq.3.5, 24.192.1-24.dq.3.6, 24.192.1-24.dq.3.7, 24.192.1-24.dq.3.8, 24.192.1-24.dq.3.9, 24.192.1-24.dq.3.10, 24.192.1-24.dq.3.11, 24.192.1-24.dq.3.12, 24.192.1-24.dq.3.13, 24.192.1-24.dq.3.14, 24.192.1-24.dq.3.15, 24.192.1-24.dq.3.16, 120.192.1-24.dq.3.1, 120.192.1-24.dq.3.2, 120.192.1-24.dq.3.3, 120.192.1-24.dq.3.4, 120.192.1-24.dq.3.5, 120.192.1-24.dq.3.6, 120.192.1-24.dq.3.7, 120.192.1-24.dq.3.8, 120.192.1-24.dq.3.9, 120.192.1-24.dq.3.10, 120.192.1-24.dq.3.11, 120.192.1-24.dq.3.12, 120.192.1-24.dq.3.13, 120.192.1-24.dq.3.14, 120.192.1-24.dq.3.15, 120.192.1-24.dq.3.16, 168.192.1-24.dq.3.1, 168.192.1-24.dq.3.2, 168.192.1-24.dq.3.3, 168.192.1-24.dq.3.4, 168.192.1-24.dq.3.5, 168.192.1-24.dq.3.6, 168.192.1-24.dq.3.7, 168.192.1-24.dq.3.8, 168.192.1-24.dq.3.9, 168.192.1-24.dq.3.10, 168.192.1-24.dq.3.11, 168.192.1-24.dq.3.12, 168.192.1-24.dq.3.13, 168.192.1-24.dq.3.14, 168.192.1-24.dq.3.15, 168.192.1-24.dq.3.16, 264.192.1-24.dq.3.1, 264.192.1-24.dq.3.2, 264.192.1-24.dq.3.3, 264.192.1-24.dq.3.4, 264.192.1-24.dq.3.5, 264.192.1-24.dq.3.6, 264.192.1-24.dq.3.7, 264.192.1-24.dq.3.8, 264.192.1-24.dq.3.9, 264.192.1-24.dq.3.10, 264.192.1-24.dq.3.11, 264.192.1-24.dq.3.12, 264.192.1-24.dq.3.13, 264.192.1-24.dq.3.14, 264.192.1-24.dq.3.15, 264.192.1-24.dq.3.16, 312.192.1-24.dq.3.1, 312.192.1-24.dq.3.2, 312.192.1-24.dq.3.3, 312.192.1-24.dq.3.4, 312.192.1-24.dq.3.5, 312.192.1-24.dq.3.6, 312.192.1-24.dq.3.7, 312.192.1-24.dq.3.8, 312.192.1-24.dq.3.9, 312.192.1-24.dq.3.10, 312.192.1-24.dq.3.11, 312.192.1-24.dq.3.12, 312.192.1-24.dq.3.13, 312.192.1-24.dq.3.14, 312.192.1-24.dq.3.15, 312.192.1-24.dq.3.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

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 $
$=$ $2 x^{2} + 8 x y + 2 y^{2} + 3 z^{2} + 3 w^{2}$
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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} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle w$

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}}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.0.c.4 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bs.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.ix.1 $24$ $2$ $2$ $1$ $0$ 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
24.192.5.de.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dk.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.eq.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.er.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fm.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fo.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fu.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fw.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.bd.1 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.be.4 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.fb.4 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.fr.3 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.baq.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bas.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbg.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bbi.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bde.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bds.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bdu.2 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.baq.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bas.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbg.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bbi.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdc.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bde.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bds.1 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.bdu.1 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.baq.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bas.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbg.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bbi.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdc.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bde.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bds.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.bdu.1 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.baq.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bas.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbg.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bbi.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdc.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bde.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bds.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.bdu.1 $312$ $2$ $2$ $5$ $?$ not computed