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Properties

Label 24.48.1.o.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

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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 $4$ are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $1^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.23

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&4\\20&19\end{bmatrix}$, $\begin{bmatrix}9&4\\10&3\end{bmatrix}$, $\begin{bmatrix}9&20\\20&9\end{bmatrix}$, $\begin{bmatrix}9&20\\22&15\end{bmatrix}$, $\begin{bmatrix}19&8\\12&23\end{bmatrix}$, $\begin{bmatrix}19&12\\16&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.o.1.1, 24.96.1-24.o.1.2, 24.96.1-24.o.1.3, 24.96.1-24.o.1.4, 24.96.1-24.o.1.5, 24.96.1-24.o.1.6, 24.96.1-24.o.1.7, 24.96.1-24.o.1.8, 24.96.1-24.o.1.9, 24.96.1-24.o.1.10, 24.96.1-24.o.1.11, 24.96.1-24.o.1.12, 120.96.1-24.o.1.1, 120.96.1-24.o.1.2, 120.96.1-24.o.1.3, 120.96.1-24.o.1.4, 120.96.1-24.o.1.5, 120.96.1-24.o.1.6, 120.96.1-24.o.1.7, 120.96.1-24.o.1.8, 120.96.1-24.o.1.9, 120.96.1-24.o.1.10, 120.96.1-24.o.1.11, 120.96.1-24.o.1.12, 168.96.1-24.o.1.1, 168.96.1-24.o.1.2, 168.96.1-24.o.1.3, 168.96.1-24.o.1.4, 168.96.1-24.o.1.5, 168.96.1-24.o.1.6, 168.96.1-24.o.1.7, 168.96.1-24.o.1.8, 168.96.1-24.o.1.9, 168.96.1-24.o.1.10, 168.96.1-24.o.1.11, 168.96.1-24.o.1.12, 264.96.1-24.o.1.1, 264.96.1-24.o.1.2, 264.96.1-24.o.1.3, 264.96.1-24.o.1.4, 264.96.1-24.o.1.5, 264.96.1-24.o.1.6, 264.96.1-24.o.1.7, 264.96.1-24.o.1.8, 264.96.1-24.o.1.9, 264.96.1-24.o.1.10, 264.96.1-24.o.1.11, 264.96.1-24.o.1.12, 312.96.1-24.o.1.1, 312.96.1-24.o.1.2, 312.96.1-24.o.1.3, 312.96.1-24.o.1.4, 312.96.1-24.o.1.5, 312.96.1-24.o.1.6, 312.96.1-24.o.1.7, 312.96.1-24.o.1.8, 312.96.1-24.o.1.9, 312.96.1-24.o.1.10, 312.96.1-24.o.1.11, 312.96.1-24.o.1.12
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} - 9x $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(-3:0:1)$, $(3:0:1)$, $(0:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^4}\cdot\frac{5670x^{2}y^{12}z^{2}+268731999x^{2}y^{8}z^{6}+1190026602045x^{2}y^{4}z^{10}+128505439098855x^{2}z^{14}+72xy^{14}z+10491039xy^{10}z^{5}+122463138276xy^{6}z^{9}+71412831316881xy^{2}z^{13}+y^{16}+224532y^{12}z^{4}+5469590772y^{8}z^{8}+6348272132754y^{4}z^{12}+282429536481z^{16}}{z^{2}y^{8}(x^{2}y^{4}+10935x^{2}z^{4}+1377xy^{2}z^{3}+54y^{4}z^{2}+6561z^{6})}$

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
$X_{\mathrm{sp}}(4)$ $4$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.k.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dr.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dx.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.a.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.dj.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.dp.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.o.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.o.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.h.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.j.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.q.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.r.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.w.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.ba.1 $24$ $2$ $2$ $3$ $0$ $2$
24.144.9.cu.2 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.bk.1 $24$ $4$ $4$ $9$ $1$ $1^{8}$
120.96.1.bm.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.bm.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.y.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.be.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bw.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.bx.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.cl.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.cr.1 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.bb.2 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.qe.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.bm.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.bm.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.q.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.w.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bo.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.bp.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.cd.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.cj.1 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.bm.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.bm.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.q.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.w.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bo.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.bp.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.cd.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.cj.1 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.bm.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.bm.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.y.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.be.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bw.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.bx.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.cl.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.cr.1 $312$ $2$ $2$ $3$ $?$ not computed