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Properties

Label 24.48.1.cg.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: $192$
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: 24.48.1.70

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&11\\0&7\end{bmatrix}$, $\begin{bmatrix}7&5\\12&5\end{bmatrix}$, $\begin{bmatrix}7&9\\0&19\end{bmatrix}$, $\begin{bmatrix}13&22\\0&23\end{bmatrix}$, $\begin{bmatrix}17&11\\0&5\end{bmatrix}$, $\begin{bmatrix}19&10\\12&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 24.96.1-24.cg.1.1, 24.96.1-24.cg.1.2, 24.96.1-24.cg.1.3, 24.96.1-24.cg.1.4, 24.96.1-24.cg.1.5, 24.96.1-24.cg.1.6, 24.96.1-24.cg.1.7, 24.96.1-24.cg.1.8, 24.96.1-24.cg.1.9, 24.96.1-24.cg.1.10, 24.96.1-24.cg.1.11, 24.96.1-24.cg.1.12, 24.96.1-24.cg.1.13, 24.96.1-24.cg.1.14, 24.96.1-24.cg.1.15, 24.96.1-24.cg.1.16, 24.96.1-24.cg.1.17, 24.96.1-24.cg.1.18, 24.96.1-24.cg.1.19, 24.96.1-24.cg.1.20, 120.96.1-24.cg.1.1, 120.96.1-24.cg.1.2, 120.96.1-24.cg.1.3, 120.96.1-24.cg.1.4, 120.96.1-24.cg.1.5, 120.96.1-24.cg.1.6, 120.96.1-24.cg.1.7, 120.96.1-24.cg.1.8, 120.96.1-24.cg.1.9, 120.96.1-24.cg.1.10, 120.96.1-24.cg.1.11, 120.96.1-24.cg.1.12, 120.96.1-24.cg.1.13, 120.96.1-24.cg.1.14, 120.96.1-24.cg.1.15, 120.96.1-24.cg.1.16, 120.96.1-24.cg.1.17, 120.96.1-24.cg.1.18, 120.96.1-24.cg.1.19, 120.96.1-24.cg.1.20, 168.96.1-24.cg.1.1, 168.96.1-24.cg.1.2, 168.96.1-24.cg.1.3, 168.96.1-24.cg.1.4, 168.96.1-24.cg.1.5, 168.96.1-24.cg.1.6, 168.96.1-24.cg.1.7, 168.96.1-24.cg.1.8, 168.96.1-24.cg.1.9, 168.96.1-24.cg.1.10, 168.96.1-24.cg.1.11, 168.96.1-24.cg.1.12, 168.96.1-24.cg.1.13, 168.96.1-24.cg.1.14, 168.96.1-24.cg.1.15, 168.96.1-24.cg.1.16, 168.96.1-24.cg.1.17, 168.96.1-24.cg.1.18, 168.96.1-24.cg.1.19, 168.96.1-24.cg.1.20, 264.96.1-24.cg.1.1, 264.96.1-24.cg.1.2, 264.96.1-24.cg.1.3, 264.96.1-24.cg.1.4, 264.96.1-24.cg.1.5, 264.96.1-24.cg.1.6, 264.96.1-24.cg.1.7, 264.96.1-24.cg.1.8, 264.96.1-24.cg.1.9, 264.96.1-24.cg.1.10, 264.96.1-24.cg.1.11, 264.96.1-24.cg.1.12, 264.96.1-24.cg.1.13, 264.96.1-24.cg.1.14, 264.96.1-24.cg.1.15, 264.96.1-24.cg.1.16, 264.96.1-24.cg.1.17, 264.96.1-24.cg.1.18, 264.96.1-24.cg.1.19, 264.96.1-24.cg.1.20, 312.96.1-24.cg.1.1, 312.96.1-24.cg.1.2, 312.96.1-24.cg.1.3, 312.96.1-24.cg.1.4, 312.96.1-24.cg.1.5, 312.96.1-24.cg.1.6, 312.96.1-24.cg.1.7, 312.96.1-24.cg.1.8, 312.96.1-24.cg.1.9, 312.96.1-24.cg.1.10, 312.96.1-24.cg.1.11, 312.96.1-24.cg.1.12, 312.96.1-24.cg.1.13, 312.96.1-24.cg.1.14, 312.96.1-24.cg.1.15, 312.96.1-24.cg.1.16, 312.96.1-24.cg.1.17, 312.96.1-24.cg.1.18, 312.96.1-24.cg.1.19, 312.96.1-24.cg.1.20
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} - 17x - 15 $
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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
$(-3:0:1)$, $(-1:0:1)$, $(5:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2}\cdot\frac{1408x^{2}y^{14}-2837390880x^{2}y^{12}z^{2}+852859782144x^{2}y^{10}z^{4}-40684227015680x^{2}y^{8}z^{6}-14198899790643200x^{2}y^{6}z^{8}-2044197213532913664x^{2}y^{4}z^{10}-114381597197698334720x^{2}y^{2}z^{12}-2271857249241291292672x^{2}z^{14}-674384xy^{14}z+28810315200xy^{12}z^{3}-3536050837248xy^{10}z^{5}-134678248503296xy^{8}z^{7}-89133784780374016xy^{6}z^{9}-10270549770837295104xy^{4}z^{11}-504857792897761673216xy^{2}z^{13}-9087432106146290204672xz^{15}-y^{16}+115447536y^{14}z^{2}-139252204320y^{12}z^{4}+3977931004160y^{10}z^{6}-1845366839198720y^{8}z^{8}-361850542509588480y^{6}z^{10}-25975269007566635008y^{4}z^{12}-769119329672038973440y^{2}z^{14}-6815581356742225690624z^{16}}{y^{2}(x^{2}y^{12}+63648x^{2}y^{10}z^{2}+95435264x^{2}y^{8}z^{4}+18043782144x^{2}y^{6}z^{6}+743109054464x^{2}y^{4}z^{8}+8589934592x^{2}y^{2}z^{10}+68719476736x^{2}z^{12}+78xy^{12}z+951216xy^{10}z^{3}+666194432xy^{8}z^{5}+87721647104xy^{6}z^{7}+2970884202496xy^{4}z^{9}-30064771072xy^{2}z^{11}-274877906944xz^{13}+2825y^{12}z^{2}+10362000y^{10}z^{4}+3154535680y^{8}z^{6}+193857198080y^{6}z^{8}+2231113568256y^{4}z^{10}-73014444032y^{2}z^{12}-343597383680z^{14})}$

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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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(3)$ $3$ $12$ $12$ $0$ $0$ full Jacobian
8.12.0.d.1 $8$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.12.0.d.1 $8$ $4$ $4$ $0$ $0$ full Jacobian
$X_0(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.p.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.es.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.cr.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cr.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cr.3 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.cr.4 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.cn.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.cn.2 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.co.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.cp.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.cq.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.cr.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.ct.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.ct.2 $24$ $2$ $2$ $3$ $0$ $2$
24.144.5.ba.1 $24$ $3$ $3$ $5$ $1$ $1^{4}$
72.144.5.f.1 $72$ $3$ $3$ $5$ $?$ not computed
72.144.9.j.1 $72$ $3$ $3$ $9$ $?$ not computed
72.144.9.l.1 $72$ $3$ $3$ $9$ $?$ not computed
120.96.1.qh.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.qh.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.qh.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.qh.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.hp.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.hp.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.hq.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.hr.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.hs.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ht.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.hv.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.hv.2 $120$ $2$ $2$ $3$ $?$ not computed
120.240.17.it.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.mqh.1 $120$ $6$ $6$ $17$ $?$ not computed
168.96.1.qh.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qh.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qh.3 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.qh.4 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.ft.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ft.2 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.fu.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.fv.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.fw.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.fx.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.fz.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.fz.2 $168$ $2$ $2$ $3$ $?$ not computed
264.96.1.qh.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qh.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qh.3 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.qh.4 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.ft.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ft.2 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.fu.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.fv.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.fw.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.fx.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.fz.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.fz.2 $264$ $2$ $2$ $3$ $?$ not computed
312.96.1.qh.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.qh.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.qh.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.qh.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.hp.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.hp.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.hq.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.hr.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.hs.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ht.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.hv.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.hv.2 $312$ $2$ $2$ $3$ $?$ not computed