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Properties

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

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $576$
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: $1$
$\Q$-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: 24.96.1.733

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}3&1\\14&1\end{bmatrix}$, $\begin{bmatrix}3&5\\16&17\end{bmatrix}$, $\begin{bmatrix}7&9\\0&17\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: $(C_2\times \GL(2,3)):D_4$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.c

Models

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

$ 0 $ $=$ $ y^{2} - 2 y z - 2 y w + z^{2} + w^{2} $
$=$ $6 x^{2} - 2 y^{2} + z^{2} + 2 z w + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} - 32 x^{3} z - 12 x^{2} y^{2} + 4 x^{2} z^{2} + 24 x y^{2} z + 9 y^{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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 2x$
$\displaystyle Z$ $=$ $\displaystyle 2w$

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{24yz^{23}+4072yz^{22}w+166008yz^{21}w^{2}+1697160yz^{20}w^{3}-16620120yz^{19}w^{4}-300279720yz^{18}w^{5}+25439880yz^{17}w^{6}+14042577272yz^{16}w^{7}+40523919600yz^{15}w^{8}-151460567280yz^{14}w^{9}-988513560144yz^{13}w^{10}-2053487662512yz^{12}w^{11}-2053487662512yz^{11}w^{12}-988513560144yz^{10}w^{13}-151460567280yz^{9}w^{14}+40523919600yz^{8}w^{15}+14042577272yz^{7}w^{16}+25439880yz^{6}w^{17}-300279720yz^{5}w^{18}-16620120yz^{4}w^{19}+1697160yz^{3}w^{20}+166008yz^{2}w^{21}+4072yzw^{22}+24yw^{23}-23z^{24}-3544z^{23}w-127276z^{22}w^{2}-927400z^{21}w^{3}+16118034z^{20}w^{4}+181625400z^{19}w^{5}-479536508z^{18}w^{6}-8805466808z^{17}w^{7}-10664661113z^{16}w^{8}+113422126352z^{15}w^{9}+460068426408z^{14}w^{10}+814917044208z^{13}w^{11}+940474615548z^{12}w^{12}+814917044208z^{11}w^{13}+460068426408z^{10}w^{14}+113422126352z^{9}w^{15}-10664661113z^{8}w^{16}-8805466808z^{7}w^{17}-479536508z^{6}w^{18}+181625400z^{5}w^{19}+16118034z^{4}w^{20}-927400z^{3}w^{21}-127276z^{2}w^{22}-3544zw^{23}-23w^{24}}{w^{2}z^{2}(z+w)^{8}(20yz^{11}+2212yz^{10}w+56724yz^{9}w^{2}+529860yz^{8}w^{3}+2178472yz^{7}w^{4}+4329032yz^{6}w^{5}+4329032yz^{5}w^{6}+2178472yz^{4}w^{7}+529860yz^{3}w^{8}+56724yz^{2}w^{9}+2212yzw^{10}+20yw^{11}-19z^{12}-1856z^{11}w-40634z^{10}w^{2}-312544z^{9}w^{3}-1037165z^{8}w^{4}-1768160z^{7}w^{5}-1993100z^{6}w^{6}-1768160z^{5}w^{7}-1037165z^{4}w^{8}-312544z^{3}w^{9}-40634z^{2}w^{10}-1856zw^{11}-19w^{12})}$

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.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bv.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.kh.1 $24$ $2$ $2$ $1$ $1$ 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.gdb.2 $24$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
24.384.17.un.2 $24$ $4$ $4$ $17$ $3$ $1^{8}\cdot2^{4}$