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Properties

Label 24.32.1.d.1
Level $24$
Index $32$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $48$
Index: $32$ $\PSL_2$-index:$32$
Genus: $1 = 1 + \frac{ 32 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $4^{2}\cdot12^{2}$ Cusp orbits $2^{2}$
Elliptic points: $0$ of order $2$ and $2$ 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: 12I1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.32.1.15

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}8&5\\9&19\end{bmatrix}$, $\begin{bmatrix}13&18\\6&11\end{bmatrix}$, $\begin{bmatrix}17&17\\6&11\end{bmatrix}$, $\begin{bmatrix}19&12\\12&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.64.1-24.d.1.1, 24.64.1-24.d.1.2, 24.64.1-24.d.1.3, 24.64.1-24.d.1.4, 24.64.1-24.d.1.5, 24.64.1-24.d.1.6, 24.64.1-24.d.1.7, 24.64.1-24.d.1.8, 120.64.1-24.d.1.1, 120.64.1-24.d.1.2, 120.64.1-24.d.1.3, 120.64.1-24.d.1.4, 120.64.1-24.d.1.5, 120.64.1-24.d.1.6, 120.64.1-24.d.1.7, 120.64.1-24.d.1.8, 168.64.1-24.d.1.1, 168.64.1-24.d.1.2, 168.64.1-24.d.1.3, 168.64.1-24.d.1.4, 168.64.1-24.d.1.5, 168.64.1-24.d.1.6, 168.64.1-24.d.1.7, 168.64.1-24.d.1.8, 264.64.1-24.d.1.1, 264.64.1-24.d.1.2, 264.64.1-24.d.1.3, 264.64.1-24.d.1.4, 264.64.1-24.d.1.5, 264.64.1-24.d.1.6, 264.64.1-24.d.1.7, 264.64.1-24.d.1.8, 312.64.1-24.d.1.1, 312.64.1-24.d.1.2, 312.64.1-24.d.1.3, 312.64.1-24.d.1.4, 312.64.1-24.d.1.5, 312.64.1-24.d.1.6, 312.64.1-24.d.1.7, 312.64.1-24.d.1.8
Cyclic 24-isogeny field degree: $12$
Cyclic 24-torsion field degree: $96$
Full 24-torsion field degree: $2304$

Jacobian

Conductor: $2^{4}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 48.2.a.a

Models

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

$ 0 $ $=$ $ 3 x^{2} - 6 x z + 2 y w $
$=$ $9 x^{2} + 6 x z - 9 y^{2} + 6 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 6 x^{3} y + 9 x^{2} y^{2} - 4 x^{2} z^{2} - 4 x y z^{2} - 6 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{3}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Maps to other modular curves

$j$-invariant map of degree 32 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2\cdot3^2\,\frac{z(3949236xyz^{4}w+944640xyz^{2}w^{3}+2304xyw^{5}-4434876xz^{6}-7676100xz^{4}w^{2}-222720xz^{2}w^{4}+768xw^{6}+1478316yz^{5}w+2229408yz^{3}w^{3}+65280yzw^{5}+9z^{7}-713340z^{5}w^{2}+49952z^{3}w^{4}+11520zw^{6})}{43200xyz^{5}w+16896xyz^{3}w^{3}+411xyzw^{5}-46656xz^{7}-97152xz^{5}w^{2}-9271xz^{3}w^{4}+17xzw^{6}+15552yz^{6}w+28736yz^{4}w^{3}+2573yz^{2}w^{5}+8yw^{7}-7872z^{6}w^{2}-434z^{4}w^{4}+279z^{2}w^{6}+2w^{8}}$

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
12.16.1.a.1 $12$ $2$ $2$ $1$ $0$ dimension zero
24.8.0.d.1 $24$ $4$ $4$ $0$ $0$ full Jacobian
24.8.0.f.1 $24$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.96.3.hu.1 $24$ $3$ $3$ $3$ $1$ $1^{2}$
24.96.5.fc.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
24.128.7.h.1 $24$ $4$ $4$ $7$ $1$ $1^{6}$
72.96.5.f.1 $72$ $3$ $3$ $5$ $?$ not computed
72.96.5.h.1 $72$ $3$ $3$ $5$ $?$ not computed
72.96.7.d.1 $72$ $3$ $3$ $7$ $?$ not computed
120.160.11.l.1 $120$ $5$ $5$ $11$ $?$ not computed
120.192.13.el.1 $120$ $6$ $6$ $13$ $?$ not computed
120.320.23.d.1 $120$ $10$ $10$ $23$ $?$ not computed
168.256.17.d.1 $168$ $8$ $8$ $17$ $?$ not computed