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Properties

Label 24.48.1.bu.1
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: 24.48.1.128

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&18\\8&7\end{bmatrix}$, $\begin{bmatrix}5&22\\16&9\end{bmatrix}$, $\begin{bmatrix}9&10\\16&13\end{bmatrix}$, $\begin{bmatrix}11&18\\16&7\end{bmatrix}$, $\begin{bmatrix}15&4\\8&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.bu.1.1, 24.96.1-24.bu.1.2, 24.96.1-24.bu.1.3, 24.96.1-24.bu.1.4, 24.96.1-24.bu.1.5, 24.96.1-24.bu.1.6, 24.96.1-24.bu.1.7, 24.96.1-24.bu.1.8, 24.96.1-24.bu.1.9, 24.96.1-24.bu.1.10, 24.96.1-24.bu.1.11, 24.96.1-24.bu.1.12, 24.96.1-24.bu.1.13, 24.96.1-24.bu.1.14, 24.96.1-24.bu.1.15, 24.96.1-24.bu.1.16, 48.96.1-24.bu.1.1, 48.96.1-24.bu.1.2, 48.96.1-24.bu.1.3, 48.96.1-24.bu.1.4, 48.96.1-24.bu.1.5, 48.96.1-24.bu.1.6, 48.96.1-24.bu.1.7, 48.96.1-24.bu.1.8, 120.96.1-24.bu.1.1, 120.96.1-24.bu.1.2, 120.96.1-24.bu.1.3, 120.96.1-24.bu.1.4, 120.96.1-24.bu.1.5, 120.96.1-24.bu.1.6, 120.96.1-24.bu.1.7, 120.96.1-24.bu.1.8, 120.96.1-24.bu.1.9, 120.96.1-24.bu.1.10, 120.96.1-24.bu.1.11, 120.96.1-24.bu.1.12, 120.96.1-24.bu.1.13, 120.96.1-24.bu.1.14, 120.96.1-24.bu.1.15, 120.96.1-24.bu.1.16, 168.96.1-24.bu.1.1, 168.96.1-24.bu.1.2, 168.96.1-24.bu.1.3, 168.96.1-24.bu.1.4, 168.96.1-24.bu.1.5, 168.96.1-24.bu.1.6, 168.96.1-24.bu.1.7, 168.96.1-24.bu.1.8, 168.96.1-24.bu.1.9, 168.96.1-24.bu.1.10, 168.96.1-24.bu.1.11, 168.96.1-24.bu.1.12, 168.96.1-24.bu.1.13, 168.96.1-24.bu.1.14, 168.96.1-24.bu.1.15, 168.96.1-24.bu.1.16, 240.96.1-24.bu.1.1, 240.96.1-24.bu.1.2, 240.96.1-24.bu.1.3, 240.96.1-24.bu.1.4, 240.96.1-24.bu.1.5, 240.96.1-24.bu.1.6, 240.96.1-24.bu.1.7, 240.96.1-24.bu.1.8, 264.96.1-24.bu.1.1, 264.96.1-24.bu.1.2, 264.96.1-24.bu.1.3, 264.96.1-24.bu.1.4, 264.96.1-24.bu.1.5, 264.96.1-24.bu.1.6, 264.96.1-24.bu.1.7, 264.96.1-24.bu.1.8, 264.96.1-24.bu.1.9, 264.96.1-24.bu.1.10, 264.96.1-24.bu.1.11, 264.96.1-24.bu.1.12, 264.96.1-24.bu.1.13, 264.96.1-24.bu.1.14, 264.96.1-24.bu.1.15, 264.96.1-24.bu.1.16, 312.96.1-24.bu.1.1, 312.96.1-24.bu.1.2, 312.96.1-24.bu.1.3, 312.96.1-24.bu.1.4, 312.96.1-24.bu.1.5, 312.96.1-24.bu.1.6, 312.96.1-24.bu.1.7, 312.96.1-24.bu.1.8, 312.96.1-24.bu.1.9, 312.96.1-24.bu.1.10, 312.96.1-24.bu.1.11, 312.96.1-24.bu.1.12, 312.96.1-24.bu.1.13, 312.96.1-24.bu.1.14, 312.96.1-24.bu.1.15, 312.96.1-24.bu.1.16
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} + 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
$(0:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^2}\cdot\frac{12921120x^{2}y^{12}z^{2}-5923931728896x^{2}y^{8}z^{6}-51499361360609280x^{2}y^{4}z^{10}-538990877234083921920x^{2}z^{14}+6192xy^{14}z+271155527424xy^{10}z^{5}+8840731536654336xy^{6}z^{9}+74881781010929811456xy^{2}z^{13}+y^{16}+9374683392y^{12}z^{4}-155483208695808y^{8}z^{8}-1645880657768349696y^{4}z^{12}+4738381338321616896z^{16}}{zy^{4}(468x^{2}y^{8}z-841487616x^{2}y^{4}z^{5}+19982861844480x^{2}z^{9}-xy^{10}+12503808xy^{6}z^{4}-1673945616384xy^{2}z^{8}-90720y^{8}z^{3}+31563343872y^{4}z^{7}-2821109907456z^{11})}$

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
8.24.0.i.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
12.24.0.c.1 $12$ $2$ $2$ $0$ $0$ 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.ca.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.ca.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cb.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cb.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cc.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cc.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cd.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cd.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.144.9.je.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.ey.1 $24$ $4$ $4$ $9$ $1$ $1^{8}$
48.96.3.bo.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.bo.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.cb.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.cb.2 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.ci.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.ci.2 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.cu.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.cu.2 $48$ $2$ $2$ $3$ $0$ $2$
120.96.1.kw.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kw.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kx.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kx.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ky.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ky.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kz.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.kz.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.240.17.fe.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.fxm.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.kw.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kw.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kx.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kx.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ky.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ky.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kz.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.kz.2 $168$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.fo.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fo.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fs.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.fs.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ft.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ft.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.gq.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.gq.2 $240$ $2$ $2$ $3$ $?$ not computed
264.96.1.kw.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kw.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kx.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kx.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ky.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ky.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kz.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.kz.2 $264$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kw.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kw.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kx.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kx.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ky.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ky.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kz.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.kz.2 $312$ $2$ $2$ $1$ $?$ dimension zero