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Properties

Label 24.48.1.bz.1
Level $24$
Index $48$
Genus $1$
Analytic rank $1$
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: $1$
$\Q$-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: 24.48.1.33

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&22\\12&19\end{bmatrix}$, $\begin{bmatrix}5&16\\0&5\end{bmatrix}$, $\begin{bmatrix}5&18\\0&19\end{bmatrix}$, $\begin{bmatrix}11&10\\6&11\end{bmatrix}$, $\begin{bmatrix}13&0\\6&5\end{bmatrix}$, $\begin{bmatrix}17&14\\0&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.bz.1.1, 24.96.1-24.bz.1.2, 24.96.1-24.bz.1.3, 24.96.1-24.bz.1.4, 24.96.1-24.bz.1.5, 24.96.1-24.bz.1.6, 24.96.1-24.bz.1.7, 24.96.1-24.bz.1.8, 24.96.1-24.bz.1.9, 24.96.1-24.bz.1.10, 24.96.1-24.bz.1.11, 24.96.1-24.bz.1.12, 24.96.1-24.bz.1.13, 24.96.1-24.bz.1.14, 24.96.1-24.bz.1.15, 24.96.1-24.bz.1.16, 24.96.1-24.bz.1.17, 24.96.1-24.bz.1.18, 24.96.1-24.bz.1.19, 24.96.1-24.bz.1.20, 120.96.1-24.bz.1.1, 120.96.1-24.bz.1.2, 120.96.1-24.bz.1.3, 120.96.1-24.bz.1.4, 120.96.1-24.bz.1.5, 120.96.1-24.bz.1.6, 120.96.1-24.bz.1.7, 120.96.1-24.bz.1.8, 120.96.1-24.bz.1.9, 120.96.1-24.bz.1.10, 120.96.1-24.bz.1.11, 120.96.1-24.bz.1.12, 120.96.1-24.bz.1.13, 120.96.1-24.bz.1.14, 120.96.1-24.bz.1.15, 120.96.1-24.bz.1.16, 120.96.1-24.bz.1.17, 120.96.1-24.bz.1.18, 120.96.1-24.bz.1.19, 120.96.1-24.bz.1.20, 168.96.1-24.bz.1.1, 168.96.1-24.bz.1.2, 168.96.1-24.bz.1.3, 168.96.1-24.bz.1.4, 168.96.1-24.bz.1.5, 168.96.1-24.bz.1.6, 168.96.1-24.bz.1.7, 168.96.1-24.bz.1.8, 168.96.1-24.bz.1.9, 168.96.1-24.bz.1.10, 168.96.1-24.bz.1.11, 168.96.1-24.bz.1.12, 168.96.1-24.bz.1.13, 168.96.1-24.bz.1.14, 168.96.1-24.bz.1.15, 168.96.1-24.bz.1.16, 168.96.1-24.bz.1.17, 168.96.1-24.bz.1.18, 168.96.1-24.bz.1.19, 168.96.1-24.bz.1.20, 264.96.1-24.bz.1.1, 264.96.1-24.bz.1.2, 264.96.1-24.bz.1.3, 264.96.1-24.bz.1.4, 264.96.1-24.bz.1.5, 264.96.1-24.bz.1.6, 264.96.1-24.bz.1.7, 264.96.1-24.bz.1.8, 264.96.1-24.bz.1.9, 264.96.1-24.bz.1.10, 264.96.1-24.bz.1.11, 264.96.1-24.bz.1.12, 264.96.1-24.bz.1.13, 264.96.1-24.bz.1.14, 264.96.1-24.bz.1.15, 264.96.1-24.bz.1.16, 264.96.1-24.bz.1.17, 264.96.1-24.bz.1.18, 264.96.1-24.bz.1.19, 264.96.1-24.bz.1.20, 312.96.1-24.bz.1.1, 312.96.1-24.bz.1.2, 312.96.1-24.bz.1.3, 312.96.1-24.bz.1.4, 312.96.1-24.bz.1.5, 312.96.1-24.bz.1.6, 312.96.1-24.bz.1.7, 312.96.1-24.bz.1.8, 312.96.1-24.bz.1.9, 312.96.1-24.bz.1.10, 312.96.1-24.bz.1.11, 312.96.1-24.bz.1.12, 312.96.1-24.bz.1.13, 312.96.1-24.bz.1.14, 312.96.1-24.bz.1.15, 312.96.1-24.bz.1.16, 312.96.1-24.bz.1.17, 312.96.1-24.bz.1.18, 312.96.1-24.bz.1.19, 312.96.1-24.bz.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.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 156x + 560 $
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Rational points

This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.

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.b.1 $24$ $4$ $4$ $0$ $0$ full Jacobian
24.24.0.cd.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.eq.1 $24$ $2$ $2$ $1$ $1$ 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.cp.1 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.cp.2 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.cp.3 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.1.cp.4 $24$ $2$ $2$ $1$ $1$ dimension zero
24.96.3.bd.1 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.96.3.be.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.bf.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.bg.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.cc.1 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.cc.2 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.cd.1 $24$ $2$ $2$ $3$ $1$ $2$
24.96.3.cd.2 $24$ $2$ $2$ $3$ $1$ $2$
24.144.5.m.1 $24$ $3$ $3$ $5$ $2$ $1^{4}$
72.144.5.d.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.d.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.h.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.1.ll.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ll.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ll.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.ll.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.ej.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ek.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.em.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.en.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fi.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fi.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fj.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fj.2 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.fj.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.ggx.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.ll.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ll.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ll.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ll.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.dl.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.dm.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.do.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.dp.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ek.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ek.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.el.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.el.2 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.ll.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ll.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ll.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.ll.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.dl.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.dm.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.do.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.dp.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ek.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ek.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.el.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.el.2 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.ll.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ll.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ll.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.ll.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.ej.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ek.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.em.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.en.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.fi.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.fi.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.fj.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.fj.2 $312$ $2$ $2$ $3$ $?$ not computed