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Properties

Label 24.48.1.o.2
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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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: $8$ (of which $2$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.22

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&4\\12&1\end{bmatrix}$, $\begin{bmatrix}7&16\\18&13\end{bmatrix}$, $\begin{bmatrix}13&8\\22&3\end{bmatrix}$, $\begin{bmatrix}15&16\\14&9\end{bmatrix}$, $\begin{bmatrix}17&4\\22&3\end{bmatrix}$, $\begin{bmatrix}23&16\\0&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.o.2.1, 24.96.1-24.o.2.2, 24.96.1-24.o.2.3, 24.96.1-24.o.2.4, 24.96.1-24.o.2.5, 24.96.1-24.o.2.6, 24.96.1-24.o.2.7, 24.96.1-24.o.2.8, 24.96.1-24.o.2.9, 24.96.1-24.o.2.10, 24.96.1-24.o.2.11, 24.96.1-24.o.2.12, 24.96.1-24.o.2.13, 24.96.1-24.o.2.14, 24.96.1-24.o.2.15, 24.96.1-24.o.2.16, 24.96.1-24.o.2.17, 24.96.1-24.o.2.18, 24.96.1-24.o.2.19, 24.96.1-24.o.2.20, 24.96.1-24.o.2.21, 24.96.1-24.o.2.22, 24.96.1-24.o.2.23, 24.96.1-24.o.2.24, 120.96.1-24.o.2.1, 120.96.1-24.o.2.2, 120.96.1-24.o.2.3, 120.96.1-24.o.2.4, 120.96.1-24.o.2.5, 120.96.1-24.o.2.6, 120.96.1-24.o.2.7, 120.96.1-24.o.2.8, 120.96.1-24.o.2.9, 120.96.1-24.o.2.10, 120.96.1-24.o.2.11, 120.96.1-24.o.2.12, 120.96.1-24.o.2.13, 120.96.1-24.o.2.14, 120.96.1-24.o.2.15, 120.96.1-24.o.2.16, 120.96.1-24.o.2.17, 120.96.1-24.o.2.18, 120.96.1-24.o.2.19, 120.96.1-24.o.2.20, 120.96.1-24.o.2.21, 120.96.1-24.o.2.22, 120.96.1-24.o.2.23, 120.96.1-24.o.2.24, 168.96.1-24.o.2.1, 168.96.1-24.o.2.2, 168.96.1-24.o.2.3, 168.96.1-24.o.2.4, 168.96.1-24.o.2.5, 168.96.1-24.o.2.6, 168.96.1-24.o.2.7, 168.96.1-24.o.2.8, 168.96.1-24.o.2.9, 168.96.1-24.o.2.10, 168.96.1-24.o.2.11, 168.96.1-24.o.2.12, 168.96.1-24.o.2.13, 168.96.1-24.o.2.14, 168.96.1-24.o.2.15, 168.96.1-24.o.2.16, 168.96.1-24.o.2.17, 168.96.1-24.o.2.18, 168.96.1-24.o.2.19, 168.96.1-24.o.2.20, 168.96.1-24.o.2.21, 168.96.1-24.o.2.22, 168.96.1-24.o.2.23, 168.96.1-24.o.2.24, 264.96.1-24.o.2.1, 264.96.1-24.o.2.2, 264.96.1-24.o.2.3, 264.96.1-24.o.2.4, 264.96.1-24.o.2.5, 264.96.1-24.o.2.6, 264.96.1-24.o.2.7, 264.96.1-24.o.2.8, 264.96.1-24.o.2.9, 264.96.1-24.o.2.10, 264.96.1-24.o.2.11, 264.96.1-24.o.2.12, 264.96.1-24.o.2.13, 264.96.1-24.o.2.14, 264.96.1-24.o.2.15, 264.96.1-24.o.2.16, 264.96.1-24.o.2.17, 264.96.1-24.o.2.18, 264.96.1-24.o.2.19, 264.96.1-24.o.2.20, 264.96.1-24.o.2.21, 264.96.1-24.o.2.22, 264.96.1-24.o.2.23, 264.96.1-24.o.2.24, 312.96.1-24.o.2.1, 312.96.1-24.o.2.2, 312.96.1-24.o.2.3, 312.96.1-24.o.2.4, 312.96.1-24.o.2.5, 312.96.1-24.o.2.6, 312.96.1-24.o.2.7, 312.96.1-24.o.2.8, 312.96.1-24.o.2.9, 312.96.1-24.o.2.10, 312.96.1-24.o.2.11, 312.96.1-24.o.2.12, 312.96.1-24.o.2.13, 312.96.1-24.o.2.14, 312.96.1-24.o.2.15, 312.96.1-24.o.2.16, 312.96.1-24.o.2.17, 312.96.1-24.o.2.18, 312.96.1-24.o.2.19, 312.96.1-24.o.2.20, 312.96.1-24.o.2.21, 312.96.1-24.o.2.22, 312.96.1-24.o.2.23, 312.96.1-24.o.2.24
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

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 36x $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(0:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^4}\cdot\frac{90720x^{2}y^{12}z^{2}-275181566976x^{2}y^{8}z^{6}+77989583391621120x^{2}y^{4}z^{10}-538990877234083921920x^{2}z^{14}-288xy^{14}z+2685705984xy^{10}z^{5}-2006436057513984xy^{6}z^{9}+74881781010929811456xy^{2}z^{13}+y^{16}-14370048y^{12}z^{4}+22403443802112y^{8}z^{8}-1664161449968664576y^{4}z^{12}+4738381338321616896z^{16}}{z^{2}y^{8}(x^{2}y^{4}-699840x^{2}z^{4}+22032xy^{2}z^{3}-216y^{4}z^{2}+1679616z^{6})}$

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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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{sp}}(4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.m.1 $24$ $2$ $2$ $0$ $0$ 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.b.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.h.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.h.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.s.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.s.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.x.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.x.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.n.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.n.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.q.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.q.2 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.r.2 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.r.3 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.u.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.u.2 $24$ $2$ $2$ $3$ $0$ $2$
24.144.9.cu.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.bk.2 $24$ $4$ $4$ $9$ $1$ $1^{8}$
120.96.1.n.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.n.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.x.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.x.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bw.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bw.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.cj.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.cj.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.bo.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bo.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bw.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bw.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bx.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bx.3 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bz.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bz.2 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.bb.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.qe.2 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.n.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.n.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.x.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.x.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bw.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bw.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.cj.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.cj.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.bg.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bg.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bo.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bo.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bp.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bp.3 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.br.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.br.2 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.n.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.n.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.x.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.x.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.bw.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.bw.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.cj.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.cj.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.bg.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bg.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bo.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bo.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bp.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bp.3 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.br.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.br.2 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.n.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.n.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.x.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.x.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.bw.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.bw.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.cj.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.cj.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.bo.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bo.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bw.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bw.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bx.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bx.3 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bz.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bz.2 $312$ $2$ $2$ $3$ $?$ not computed