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Properties

Label 24.48.1.dj.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: 24.48.1.190

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}7&18\\0&23\end{bmatrix}$, $\begin{bmatrix}7&18\\16&11\end{bmatrix}$, $\begin{bmatrix}9&19\\16&15\end{bmatrix}$, $\begin{bmatrix}13&7\\8&19\end{bmatrix}$, $\begin{bmatrix}17&6\\16&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.dj.1.1, 24.96.1-24.dj.1.2, 24.96.1-24.dj.1.3, 24.96.1-24.dj.1.4, 24.96.1-24.dj.1.5, 24.96.1-24.dj.1.6, 48.96.1-24.dj.1.1, 48.96.1-24.dj.1.2, 48.96.1-24.dj.1.3, 48.96.1-24.dj.1.4, 48.96.1-24.dj.1.5, 48.96.1-24.dj.1.6, 120.96.1-24.dj.1.1, 120.96.1-24.dj.1.2, 120.96.1-24.dj.1.3, 120.96.1-24.dj.1.4, 120.96.1-24.dj.1.5, 120.96.1-24.dj.1.6, 168.96.1-24.dj.1.1, 168.96.1-24.dj.1.2, 168.96.1-24.dj.1.3, 168.96.1-24.dj.1.4, 168.96.1-24.dj.1.5, 168.96.1-24.dj.1.6, 240.96.1-24.dj.1.1, 240.96.1-24.dj.1.2, 240.96.1-24.dj.1.3, 240.96.1-24.dj.1.4, 240.96.1-24.dj.1.5, 240.96.1-24.dj.1.6, 264.96.1-24.dj.1.1, 264.96.1-24.dj.1.2, 264.96.1-24.dj.1.3, 264.96.1-24.dj.1.4, 264.96.1-24.dj.1.5, 264.96.1-24.dj.1.6, 312.96.1-24.dj.1.1, 312.96.1-24.dj.1.2, 312.96.1-24.dj.1.3, 312.96.1-24.dj.1.4, 312.96.1-24.dj.1.5, 312.96.1-24.dj.1.6
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
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
$(0:1:0)$, $(-3:0:1)$, $(0:0:1)$, $(3:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^2}{3^2}\cdot\frac{807570x^{2}y^{12}z^{2}+5785089579x^{2}y^{8}z^{6}-785817891855x^{2}y^{4}z^{10}+128505439098855x^{2}z^{14}-1548xy^{14}z+1059201279xy^{10}z^{5}-539595430704xy^{6}z^{9}+71412831316881xy^{2}z^{13}+y^{16}-146479428y^{12}z^{4}-37959767748y^{8}z^{8}+6278536444734y^{4}z^{12}+282429536481z^{16}}{zy^{4}(117x^{2}y^{8}z+3287061x^{2}y^{4}z^{5}+1219657095x^{2}z^{9}+xy^{10}+195372xy^{6}z^{4}+408678129xy^{2}z^{8}+5670y^{8}z^{3}+30823578y^{4}z^{7}+43046721z^{11})}$

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.q.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
12.24.0.e.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.di.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dj.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.m.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.da.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.db.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.cu.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cu.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.uw.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.hy.1 $24$ $4$ $4$ $9$ $1$ $1^{8}$
48.96.3.fi.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.fm.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.fw.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.ga.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.gf.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.gi.1 $48$ $2$ $2$ $3$ $0$ $2$
120.96.1.qk.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.qk.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.240.17.nm.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.pwf.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.qk.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qk.2 $168$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.qr.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.qz.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.rk.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.rl.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.su.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.sy.1 $240$ $2$ $2$ $3$ $?$ not computed
264.96.1.qk.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qk.2 $264$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.qk.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.qk.2 $312$ $2$ $2$ $1$ $?$ dimension zero