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Properties

Label 24.48.1.hr.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 $2^{2}\cdot4$
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: 24.48.1.409

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&9\\0&7\end{bmatrix}$, $\begin{bmatrix}13&16\\20&9\end{bmatrix}$, $\begin{bmatrix}17&13\\4&7\end{bmatrix}$, $\begin{bmatrix}21&20\\2&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.hr.1.1, 24.96.1-24.hr.1.2, 48.96.1-24.hr.1.1, 48.96.1-24.hr.1.2, 48.96.1-24.hr.1.3, 48.96.1-24.hr.1.4, 120.96.1-24.hr.1.1, 120.96.1-24.hr.1.2, 168.96.1-24.hr.1.1, 168.96.1-24.hr.1.2, 240.96.1-24.hr.1.1, 240.96.1-24.hr.1.2, 240.96.1-24.hr.1.3, 240.96.1-24.hr.1.4, 264.96.1-24.hr.1.1, 264.96.1-24.hr.1.2, 312.96.1-24.hr.1.1, 312.96.1-24.hr.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 $ $=$ $ y^{2} - 2 y z - 2 z^{2} + 2 w^{2} $
$=$ $6 x^{2} - y^{2} - y z - z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} - 8 x^{2} y^{2} - 6 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^6\cdot3^3\,\frac{1347840yz^{11}-2246400yz^{9}w^{2}+1355904yz^{7}w^{4}-357504yz^{5}w^{6}+38672yz^{3}w^{8}-1200yzw^{10}+986688z^{12}-2422656z^{10}w^{2}+2159856z^{8}w^{4}-871616z^{6}w^{6}+158284z^{4}w^{8}-10680z^{2}w^{10}+125w^{12}}{w^{8}(36yz^{3}-12yzw^{2}+27z^{4}-30z^{2}w^{2}+4w^{4})}$

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.bb.1 $8$ $2$ $2$ $1$ $0$ dimension zero
12.24.0.n.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dd.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.ed.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.ek.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.bc.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.bq.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.brv.1 $24$ $3$ $3$ $9$ $2$ $1^{8}$
24.192.9.ol.1 $24$ $4$ $4$ $9$ $0$ $1^{8}$
48.96.3.lo.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.lq.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.pg.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.pi.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
120.240.17.bbn.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.zex.1 $120$ $6$ $6$ $17$ $?$ not computed
240.96.3.bos.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bou.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bpy.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bqa.1 $240$ $2$ $2$ $3$ $?$ not computed