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Properties

Label 24.48.1.dq.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: $192$
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: 24.48.1.74

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&3\\6&23\end{bmatrix}$, $\begin{bmatrix}13&15\\10&5\end{bmatrix}$, $\begin{bmatrix}13&18\\10&7\end{bmatrix}$, $\begin{bmatrix}19&12\\4&11\end{bmatrix}$, $\begin{bmatrix}23&21\\18&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.dq.1.1, 24.96.1-24.dq.1.2, 24.96.1-24.dq.1.3, 24.96.1-24.dq.1.4, 24.96.1-24.dq.1.5, 24.96.1-24.dq.1.6, 24.96.1-24.dq.1.7, 24.96.1-24.dq.1.8, 24.96.1-24.dq.1.9, 24.96.1-24.dq.1.10, 24.96.1-24.dq.1.11, 24.96.1-24.dq.1.12, 48.96.1-24.dq.1.1, 48.96.1-24.dq.1.2, 48.96.1-24.dq.1.3, 48.96.1-24.dq.1.4, 48.96.1-24.dq.1.5, 48.96.1-24.dq.1.6, 48.96.1-24.dq.1.7, 48.96.1-24.dq.1.8, 120.96.1-24.dq.1.1, 120.96.1-24.dq.1.2, 120.96.1-24.dq.1.3, 120.96.1-24.dq.1.4, 120.96.1-24.dq.1.5, 120.96.1-24.dq.1.6, 120.96.1-24.dq.1.7, 120.96.1-24.dq.1.8, 120.96.1-24.dq.1.9, 120.96.1-24.dq.1.10, 120.96.1-24.dq.1.11, 120.96.1-24.dq.1.12, 168.96.1-24.dq.1.1, 168.96.1-24.dq.1.2, 168.96.1-24.dq.1.3, 168.96.1-24.dq.1.4, 168.96.1-24.dq.1.5, 168.96.1-24.dq.1.6, 168.96.1-24.dq.1.7, 168.96.1-24.dq.1.8, 168.96.1-24.dq.1.9, 168.96.1-24.dq.1.10, 168.96.1-24.dq.1.11, 168.96.1-24.dq.1.12, 240.96.1-24.dq.1.1, 240.96.1-24.dq.1.2, 240.96.1-24.dq.1.3, 240.96.1-24.dq.1.4, 240.96.1-24.dq.1.5, 240.96.1-24.dq.1.6, 240.96.1-24.dq.1.7, 240.96.1-24.dq.1.8, 264.96.1-24.dq.1.1, 264.96.1-24.dq.1.2, 264.96.1-24.dq.1.3, 264.96.1-24.dq.1.4, 264.96.1-24.dq.1.5, 264.96.1-24.dq.1.6, 264.96.1-24.dq.1.7, 264.96.1-24.dq.1.8, 264.96.1-24.dq.1.9, 264.96.1-24.dq.1.10, 264.96.1-24.dq.1.11, 264.96.1-24.dq.1.12, 312.96.1-24.dq.1.1, 312.96.1-24.dq.1.2, 312.96.1-24.dq.1.3, 312.96.1-24.dq.1.4, 312.96.1-24.dq.1.5, 312.96.1-24.dq.1.6, 312.96.1-24.dq.1.7, 312.96.1-24.dq.1.8, 312.96.1-24.dq.1.9, 312.96.1-24.dq.1.10, 312.96.1-24.dq.1.11, 312.96.1-24.dq.1.12
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.b

Models

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

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

$ 0 $ $=$ $ 9 x^{4} + 10 x^{2} z^{2} - 2 y^{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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 3^3\,\frac{(z-w)^{3}(9z^{3}-27z^{2}w+3zw^{2}+31w^{3})^{3}}{w^{6}(z-2w)^{2}(z+w)(3z-5w)^{3}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(3)$ $3$ $12$ $12$ $0$ $0$ full Jacobian
8.12.0.h.1 $8$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.12.0.h.1 $8$ $4$ $4$ $0$ $0$ full Jacobian
12.24.0.f.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.y.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.es.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
24.96.3.dn.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.dn.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.do.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.do.2 $24$ $2$ $2$ $3$ $0$ $2$
24.144.5.ch.1 $24$ $3$ $3$ $5$ $2$ $1^{4}$
48.96.3.gv.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.gw.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.gx.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.gy.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.5.ey.1 $48$ $2$ $2$ $5$ $0$ $4$
48.96.5.ey.2 $48$ $2$ $2$ $5$ $0$ $4$
48.96.5.fb.1 $48$ $2$ $2$ $5$ $0$ $4$
48.96.5.fb.2 $48$ $2$ $2$ $5$ $0$ $4$
72.144.5.n.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.z.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.bc.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.3.il.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.il.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.im.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.im.2 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.nz.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.qou.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.3.gp.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.gp.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.gq.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.gq.2 $168$ $2$ $2$ $3$ $?$ not computed
240.96.3.tt.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.tu.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.tv.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.tw.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.5.ma.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ma.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.md.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.md.2 $240$ $2$ $2$ $5$ $?$ not computed
264.96.3.gp.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.gp.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.gq.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.gq.2 $264$ $2$ $2$ $3$ $?$ not computed
312.96.3.il.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.il.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.im.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.im.2 $312$ $2$ $2$ $3$ $?$ not computed