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Properties

Label 24.48.1.ha.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: $8$ Newform level: $64$
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 $4^{4}\cdot8^{4}$ Cusp orbits $4^{2}$
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: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.364

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&10\\6&13\end{bmatrix}$, $\begin{bmatrix}11&5\\8&21\end{bmatrix}$, $\begin{bmatrix}15&14\\22&13\end{bmatrix}$, $\begin{bmatrix}19&17\\12&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.ha.1.1, 24.96.1-24.ha.1.2, 48.96.1-24.ha.1.1, 48.96.1-24.ha.1.2, 48.96.1-24.ha.1.3, 48.96.1-24.ha.1.4, 120.96.1-24.ha.1.1, 120.96.1-24.ha.1.2, 168.96.1-24.ha.1.1, 168.96.1-24.ha.1.2, 240.96.1-24.ha.1.1, 240.96.1-24.ha.1.2, 240.96.1-24.ha.1.3, 240.96.1-24.ha.1.4, 264.96.1-24.ha.1.1, 264.96.1-24.ha.1.2, 312.96.1-24.ha.1.1, 312.96.1-24.ha.1.2
Cyclic 24-isogeny field degree: $16$
Cyclic 24-torsion field degree: $128$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ x^{2} - 2 x z - 2 z^{2} - w^{2} $
$=$ $x^{2} + x z - 12 y^{2} + z^{2}$
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Singular plane model Singular plane model

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

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^8\cdot3^3\,\frac{z^{3}(3z^{2}+w^{2})(7020xz^{6}+3510xz^{4}w^{2}+438xz^{2}w^{4}+8xw^{6}+5139z^{7}+4596z^{5}w^{2}+1165z^{3}w^{4}+76zw^{6})}{w^{4}(4536xz^{7}+2268xz^{5}w^{2}+324xz^{3}w^{4}+12xzw^{6}+3321z^{8}+2970z^{6}w^{2}+783z^{4}w^{4}+66z^{2}w^{6}+w^{8})}$

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
8.24.1.ba.1 $8$ $2$ $2$ $1$ $0$ dimension zero
12.24.0.k.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.ct.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.ds.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.ek.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.y.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.bg.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
24.144.9.bpy.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.nu.1 $24$ $4$ $4$ $9$ $1$ $1^{8}$
48.96.3.kx.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.kz.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.nz.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.ob.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
120.240.17.zq.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.zbu.1 $120$ $6$ $6$ $17$ $?$ not computed
240.96.3.bmv.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bmx.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bob.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bod.1 $240$ $2$ $2$ $3$ $?$ not computed