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Properties

Label 24.48.1.if.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $576$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ Cusp orbits $2^{4}$
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: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.47

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&17\\0&13\end{bmatrix}$, $\begin{bmatrix}13&0\\6&7\end{bmatrix}$, $\begin{bmatrix}23&20\\0&23\end{bmatrix}$, $\begin{bmatrix}23&23\\6&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.if.1.1, 24.96.1-24.if.1.2, 24.96.1-24.if.1.3, 24.96.1-24.if.1.4, 24.96.1-24.if.1.5, 24.96.1-24.if.1.6, 24.96.1-24.if.1.7, 24.96.1-24.if.1.8, 120.96.1-24.if.1.1, 120.96.1-24.if.1.2, 120.96.1-24.if.1.3, 120.96.1-24.if.1.4, 120.96.1-24.if.1.5, 120.96.1-24.if.1.6, 120.96.1-24.if.1.7, 120.96.1-24.if.1.8, 168.96.1-24.if.1.1, 168.96.1-24.if.1.2, 168.96.1-24.if.1.3, 168.96.1-24.if.1.4, 168.96.1-24.if.1.5, 168.96.1-24.if.1.6, 168.96.1-24.if.1.7, 168.96.1-24.if.1.8, 264.96.1-24.if.1.1, 264.96.1-24.if.1.2, 264.96.1-24.if.1.3, 264.96.1-24.if.1.4, 264.96.1-24.if.1.5, 264.96.1-24.if.1.6, 264.96.1-24.if.1.7, 264.96.1-24.if.1.8, 312.96.1-24.if.1.1, 312.96.1-24.if.1.2, 312.96.1-24.if.1.3, 312.96.1-24.if.1.4, 312.96.1-24.if.1.5, 312.96.1-24.if.1.6, 312.96.1-24.if.1.7, 312.96.1-24.if.1.8
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $1536$

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^{2} - 2 y z $
$=$ $3 x^{2} - 7 y^{2} + 2 y z + y w - z^{2} - z w - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 28 x^{4} - 2 x^{2} y z - 16 x^{2} z^{2} + y^{2} z^{2} + y z^{3} + 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:

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

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{7^2}\cdot\frac{9448990857216yz^{11}+1574903079936yz^{10}w-10236058642944yz^{9}w^{2}-2842888534272yz^{8}w^{3}+2762569259136yz^{7}w^{4}+904262885568yz^{6}w^{5}-204328601568yz^{5}w^{6}-86704187184yz^{4}w^{7}+3203128800yz^{3}w^{8}+2866505760yz^{2}w^{9}+54686664yzw^{10}-25019280yw^{11}-1349860517888z^{12}-1574903079936z^{11}w-337805056512z^{10}w^{2}+1343061764352z^{9}w^{3}+1324551125376z^{8}w^{4}+18659654208z^{7}w^{5}-298304772192z^{6}w^{6}-52388950608z^{5}w^{7}+16408031328z^{4}w^{8}+4263770448z^{3}w^{9}-82382832z^{2}w^{10}-85917024zw^{11}-6205977w^{12}}{z^{2}(72yz^{9}+732yz^{8}w+3642yz^{7}w^{2}+12075yz^{6}w^{3}+29946yz^{5}w^{4}+59052yz^{4}w^{5}-1647114yz^{3}w^{6}-1027437yz^{2}w^{7}+57474yzw^{8}+55071yw^{9}-72z^{10}-732z^{9}w-3630z^{8}w^{2}-11951z^{7}w^{3}-29316z^{6}w^{4}-56910z^{5}w^{5}+158011z^{4}w^{6}+291345z^{3}w^{7}+324915z^{2}w^{8}+118387zw^{9}+3078w^{10})}$

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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
6.24.0.b.1 $6$ $2$ $2$ $0$ $0$ full Jacobian
24.12.0.n.1 $24$ $4$ $4$ $0$ $0$ full Jacobian
24.24.0.bw.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.er.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
24.144.5.gd.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
72.144.5.z.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.cc.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.cd.1 $72$ $3$ $3$ $9$ $?$ not computed
120.240.17.bpr.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.bkev.1 $120$ $6$ $6$ $17$ $?$ not computed