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Properties

Label 24.48.1.bc.2
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: $288$
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 $2^{4}$
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: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.105

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&10\\12&23\end{bmatrix}$, $\begin{bmatrix}5&18\\8&13\end{bmatrix}$, $\begin{bmatrix}15&8\\8&21\end{bmatrix}$, $\begin{bmatrix}15&10\\20&7\end{bmatrix}$, $\begin{bmatrix}21&20\\20&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.bc.2.1, 24.96.1-24.bc.2.2, 24.96.1-24.bc.2.3, 24.96.1-24.bc.2.4, 24.96.1-24.bc.2.5, 24.96.1-24.bc.2.6, 24.96.1-24.bc.2.7, 24.96.1-24.bc.2.8, 24.96.1-24.bc.2.9, 24.96.1-24.bc.2.10, 24.96.1-24.bc.2.11, 24.96.1-24.bc.2.12, 24.96.1-24.bc.2.13, 24.96.1-24.bc.2.14, 24.96.1-24.bc.2.15, 24.96.1-24.bc.2.16, 120.96.1-24.bc.2.1, 120.96.1-24.bc.2.2, 120.96.1-24.bc.2.3, 120.96.1-24.bc.2.4, 120.96.1-24.bc.2.5, 120.96.1-24.bc.2.6, 120.96.1-24.bc.2.7, 120.96.1-24.bc.2.8, 120.96.1-24.bc.2.9, 120.96.1-24.bc.2.10, 120.96.1-24.bc.2.11, 120.96.1-24.bc.2.12, 120.96.1-24.bc.2.13, 120.96.1-24.bc.2.14, 120.96.1-24.bc.2.15, 120.96.1-24.bc.2.16, 168.96.1-24.bc.2.1, 168.96.1-24.bc.2.2, 168.96.1-24.bc.2.3, 168.96.1-24.bc.2.4, 168.96.1-24.bc.2.5, 168.96.1-24.bc.2.6, 168.96.1-24.bc.2.7, 168.96.1-24.bc.2.8, 168.96.1-24.bc.2.9, 168.96.1-24.bc.2.10, 168.96.1-24.bc.2.11, 168.96.1-24.bc.2.12, 168.96.1-24.bc.2.13, 168.96.1-24.bc.2.14, 168.96.1-24.bc.2.15, 168.96.1-24.bc.2.16, 264.96.1-24.bc.2.1, 264.96.1-24.bc.2.2, 264.96.1-24.bc.2.3, 264.96.1-24.bc.2.4, 264.96.1-24.bc.2.5, 264.96.1-24.bc.2.6, 264.96.1-24.bc.2.7, 264.96.1-24.bc.2.8, 264.96.1-24.bc.2.9, 264.96.1-24.bc.2.10, 264.96.1-24.bc.2.11, 264.96.1-24.bc.2.12, 264.96.1-24.bc.2.13, 264.96.1-24.bc.2.14, 264.96.1-24.bc.2.15, 264.96.1-24.bc.2.16, 312.96.1-24.bc.2.1, 312.96.1-24.bc.2.2, 312.96.1-24.bc.2.3, 312.96.1-24.bc.2.4, 312.96.1-24.bc.2.5, 312.96.1-24.bc.2.6, 312.96.1-24.bc.2.7, 312.96.1-24.bc.2.8, 312.96.1-24.bc.2.9, 312.96.1-24.bc.2.10, 312.96.1-24.bc.2.11, 312.96.1-24.bc.2.12, 312.96.1-24.bc.2.13, 312.96.1-24.bc.2.14, 312.96.1-24.bc.2.15, 312.96.1-24.bc.2.16
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

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

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

$ 0 $ $=$ $ 18 x^{4} + x^{2} y^{2} - 9 x^{2} z^{2} + 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 3z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^4}{3^2}\cdot\frac{45927y^{2}z^{10}-45927y^{2}z^{8}w^{2}+1458y^{2}z^{6}w^{4}+486y^{2}z^{4}w^{6}-1701y^{2}z^{2}w^{8}+189y^{2}w^{10}+23328z^{12}-46656z^{10}w^{2}+20655z^{8}w^{4}-1728z^{6}w^{6}+432z^{4}w^{8}+180z^{2}w^{10}-31w^{12}}{w^{4}z^{4}(9y^{2}z^{2}+3y^{2}w^{2}-w^{4})}$

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
8.24.0.d.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.h.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.c.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
24.96.1.b.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.h.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bb.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bd.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bt.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.bv.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.ca.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cb.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.gq.1 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
24.192.9.dq.2 $24$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
120.96.1.fr.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ft.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.gh.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.gj.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.id.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.if.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.it.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.iv.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.240.17.dc.2 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.eze.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.fr.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ft.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.gh.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.gj.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.id.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.if.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.it.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.iv.2 $168$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.fr.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ft.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.gh.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.gj.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.id.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.if.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.it.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.iv.2 $264$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.fr.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ft.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.gh.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.gj.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.id.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.if.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.it.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.iv.2 $312$ $2$ $2$ $1$ $?$ dimension zero