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Properties

Label 24.48.1.by.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: $12$ Newform level: $576$
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 $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ Cusp orbits $1^{4}\cdot2^{2}$
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: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.34

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&18\\0&13\end{bmatrix}$, $\begin{bmatrix}1&20\\0&19\end{bmatrix}$, $\begin{bmatrix}5&0\\0&5\end{bmatrix}$, $\begin{bmatrix}11&20\\6&23\end{bmatrix}$, $\begin{bmatrix}17&14\\0&19\end{bmatrix}$, $\begin{bmatrix}23&12\\12&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.by.1.1, 24.96.1-24.by.1.2, 24.96.1-24.by.1.3, 24.96.1-24.by.1.4, 24.96.1-24.by.1.5, 24.96.1-24.by.1.6, 24.96.1-24.by.1.7, 24.96.1-24.by.1.8, 24.96.1-24.by.1.9, 24.96.1-24.by.1.10, 24.96.1-24.by.1.11, 24.96.1-24.by.1.12, 24.96.1-24.by.1.13, 24.96.1-24.by.1.14, 24.96.1-24.by.1.15, 24.96.1-24.by.1.16, 24.96.1-24.by.1.17, 24.96.1-24.by.1.18, 24.96.1-24.by.1.19, 24.96.1-24.by.1.20, 120.96.1-24.by.1.1, 120.96.1-24.by.1.2, 120.96.1-24.by.1.3, 120.96.1-24.by.1.4, 120.96.1-24.by.1.5, 120.96.1-24.by.1.6, 120.96.1-24.by.1.7, 120.96.1-24.by.1.8, 120.96.1-24.by.1.9, 120.96.1-24.by.1.10, 120.96.1-24.by.1.11, 120.96.1-24.by.1.12, 120.96.1-24.by.1.13, 120.96.1-24.by.1.14, 120.96.1-24.by.1.15, 120.96.1-24.by.1.16, 120.96.1-24.by.1.17, 120.96.1-24.by.1.18, 120.96.1-24.by.1.19, 120.96.1-24.by.1.20, 168.96.1-24.by.1.1, 168.96.1-24.by.1.2, 168.96.1-24.by.1.3, 168.96.1-24.by.1.4, 168.96.1-24.by.1.5, 168.96.1-24.by.1.6, 168.96.1-24.by.1.7, 168.96.1-24.by.1.8, 168.96.1-24.by.1.9, 168.96.1-24.by.1.10, 168.96.1-24.by.1.11, 168.96.1-24.by.1.12, 168.96.1-24.by.1.13, 168.96.1-24.by.1.14, 168.96.1-24.by.1.15, 168.96.1-24.by.1.16, 168.96.1-24.by.1.17, 168.96.1-24.by.1.18, 168.96.1-24.by.1.19, 168.96.1-24.by.1.20, 264.96.1-24.by.1.1, 264.96.1-24.by.1.2, 264.96.1-24.by.1.3, 264.96.1-24.by.1.4, 264.96.1-24.by.1.5, 264.96.1-24.by.1.6, 264.96.1-24.by.1.7, 264.96.1-24.by.1.8, 264.96.1-24.by.1.9, 264.96.1-24.by.1.10, 264.96.1-24.by.1.11, 264.96.1-24.by.1.12, 264.96.1-24.by.1.13, 264.96.1-24.by.1.14, 264.96.1-24.by.1.15, 264.96.1-24.by.1.16, 264.96.1-24.by.1.17, 264.96.1-24.by.1.18, 264.96.1-24.by.1.19, 264.96.1-24.by.1.20, 312.96.1-24.by.1.1, 312.96.1-24.by.1.2, 312.96.1-24.by.1.3, 312.96.1-24.by.1.4, 312.96.1-24.by.1.5, 312.96.1-24.by.1.6, 312.96.1-24.by.1.7, 312.96.1-24.by.1.8, 312.96.1-24.by.1.9, 312.96.1-24.by.1.10, 312.96.1-24.by.1.11, 312.96.1-24.by.1.12, 312.96.1-24.by.1.13, 312.96.1-24.by.1.14, 312.96.1-24.by.1.15, 312.96.1-24.by.1.16, 312.96.1-24.by.1.17, 312.96.1-24.by.1.18, 312.96.1-24.by.1.19, 312.96.1-24.by.1.20
Cyclic 24-isogeny field degree: $4$
Cyclic 24-torsion field degree: $32$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 156x - 560 $
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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
$(-10:0:1)$, $(-4:0:1)$, $(0:1:0)$, $(14:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^6\cdot3^6}\cdot\frac{96x^{2}y^{14}+3071520x^{2}y^{12}z^{2}+19167777792x^{2}y^{10}z^{4}+73205492290560x^{2}y^{8}z^{6}+168881565187768320x^{2}y^{6}z^{8}+266227264601441501184x^{2}y^{4}z^{10}+247916413700751470100480x^{2}y^{2}z^{12}+134702959110560452221861888x^{2}z^{14}+4368xy^{14}z+69361920xy^{12}z^{3}+383703236352xy^{10}z^{5}+1313691284017152xy^{8}z^{7}+2857875029712175104xy^{6}z^{9}+4206511845249122304000xy^{4}z^{11}+3838872231829646418640896xy^{2}z^{13}+1872592528727280463160279040xz^{15}+y^{16}+124800y^{14}z^{2}+1014923520y^{12}z^{4}+4497071063040y^{10}z^{6}+11992345692291072y^{8}z^{8}+22043062823581384704y^{6}z^{10}+25772226110498737225728y^{4}z^{12}+18749649047391718128746496y^{2}z^{14}+5335126223420150253550043136z^{16}}{z^{4}y^{4}(72x^{2}y^{6}+1191024x^{2}y^{4}z^{2}+3630210048x^{2}y^{2}z^{4}+2971852084224x^{2}z^{6}+2520xy^{6}z+23556096xy^{4}z^{3}+57717764352xy^{2}z^{5}+41609194352640xz^{7}+y^{8}+56448y^{6}z^{2}+255633408y^{4}z^{4}+337961134080y^{2}z^{6}+118906735104000z^{8})}$

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
$X_{\pm1}(2,6)$ $6$ $2$ $2$ $0$ $0$ full Jacobian
24.12.0.a.1 $24$ $4$ $4$ $0$ $0$ full Jacobian
24.24.0.cc.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.er.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.cn.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cn.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cn.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cn.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.bb.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.bc.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.bf.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.bh.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.bz.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.bz.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.ca.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.ca.2 $24$ $2$ $2$ $3$ $0$ $2$
24.144.5.p.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
72.144.5.c.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.c.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.g.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.1.lj.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.lj.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.lj.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.lj.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.ed.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ee.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.eg.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.eh.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ff.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ff.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fg.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fg.2 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.fi.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.ggw.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.lj.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.lj.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.lj.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.lj.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.df.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.dg.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.di.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.dj.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.eh.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.eh.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ei.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ei.2 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.lj.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.lj.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.lj.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.lj.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.df.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.dg.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.di.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.dj.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.eh.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.eh.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ei.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ei.2 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.lj.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.lj.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.lj.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.lj.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.ed.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ee.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.eg.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.eh.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ff.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ff.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.fg.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.fg.2 $312$ $2$ $2$ $3$ $?$ not computed