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Properties

Label 24.96.1.dv.2
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $288$
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 $4^{8}\cdot8^{8}$ Cusp orbits $2^{2}\cdot4^{3}$
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: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.730

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&8\\16&21\end{bmatrix}$, $\begin{bmatrix}1&20\\10&7\end{bmatrix}$, $\begin{bmatrix}19&9\\6&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $C_2^3.\GL(2,\mathbb{Z}/4)$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x w + y^{2} - y z + z^{2} $
$=$ $2 x^{2} - 2 y^{2} + 2 y z + z^{2} - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 2 x^{3} y + 3 x^{2} y^{2} + 2 x^{2} z^{2} - 2 x y^{3} + 4 x y z^{2} + y^{4} - 4 y^{2} z^{2} - 4 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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -3^3\,\frac{314928xz^{22}w-20015424xz^{20}w^{3}+361490688xz^{18}w^{5}-2510466048xz^{16}w^{7}+5119303680xz^{14}w^{9}+15055552512xz^{12}w^{11}-66790490112xz^{10}w^{13}+5503451136xz^{8}w^{15}+223530188800xz^{6}w^{17}-159991726080xz^{4}w^{19}-236810403840xz^{2}w^{21}+232532213760xw^{23}+19683z^{24}-3464208z^{22}w^{2}+100706976z^{20}w^{4}-1029977856z^{18}w^{6}+3793132800z^{16}w^{8}+1352982528z^{14}w^{10}-36546232320z^{12}w^{12}+44841959424z^{10}w^{14}+95589040128z^{8}w^{16}-192314081280z^{6}w^{18}-36194746368z^{4}w^{20}+221408919552z^{2}w^{22}-96317997056w^{24}}{w^{4}(3z^{2}-4w^{2})^{4}(9720xz^{10}w-387504xz^{8}w^{3}+3929472xz^{6}w^{5}-15252480xz^{4}w^{7}+24784896xz^{2}w^{9}-14192640xw^{11}+729z^{12}-84564z^{10}w^{2}+1403892z^{8}w^{4}-7781184z^{6}w^{6}+18050688z^{4}w^{8}-17793024z^{2}w^{10}+5878784w^{12})}$

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
8.48.0.q.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bv.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.kd.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
24.288.17.gcz.1 $24$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
24.384.17.um.1 $24$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$