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Properties

Label 48.48.1.bz.2
Level $48$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $48$ $\SL_2$-level: $16$ 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 $2^{4}\cdot4^{2}\cdot16^{2}$ 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: 16G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.53

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&30\\32&37\end{bmatrix}$, $\begin{bmatrix}11&16\\0&7\end{bmatrix}$, $\begin{bmatrix}17&9\\0&47\end{bmatrix}$, $\begin{bmatrix}31&23\\24&41\end{bmatrix}$, $\begin{bmatrix}35&15\\20&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.bz.2.1, 48.96.1-48.bz.2.2, 48.96.1-48.bz.2.3, 48.96.1-48.bz.2.4, 48.96.1-48.bz.2.5, 48.96.1-48.bz.2.6, 48.96.1-48.bz.2.7, 48.96.1-48.bz.2.8, 48.96.1-48.bz.2.9, 48.96.1-48.bz.2.10, 48.96.1-48.bz.2.11, 48.96.1-48.bz.2.12, 48.96.1-48.bz.2.13, 48.96.1-48.bz.2.14, 48.96.1-48.bz.2.15, 48.96.1-48.bz.2.16, 240.96.1-48.bz.2.1, 240.96.1-48.bz.2.2, 240.96.1-48.bz.2.3, 240.96.1-48.bz.2.4, 240.96.1-48.bz.2.5, 240.96.1-48.bz.2.6, 240.96.1-48.bz.2.7, 240.96.1-48.bz.2.8, 240.96.1-48.bz.2.9, 240.96.1-48.bz.2.10, 240.96.1-48.bz.2.11, 240.96.1-48.bz.2.12, 240.96.1-48.bz.2.13, 240.96.1-48.bz.2.14, 240.96.1-48.bz.2.15, 240.96.1-48.bz.2.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

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} - 99x - 378 $
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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
$(-6:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3}\cdot\frac{2232x^{2}y^{14}+69801223986x^{2}y^{12}z^{2}+8630998350150888x^{2}y^{10}z^{4}+141956909380425624561x^{2}y^{8}z^{6}+717926339135135488089168x^{2}y^{6}z^{8}+1475046256649715216484595289x^{2}y^{4}z^{10}+1303320875901571860504901151148x^{2}y^{2}z^{12}+411715752770322226985025083340825x^{2}z^{14}+1771884xy^{14}z+4867802805816xy^{12}z^{3}+266672882927098719xy^{10}z^{5}+2969872480609148627694xy^{8}z^{7}+11869594400810698466733144xy^{6}z^{9}+20790582672642410716129086408xy^{4}z^{11}+16356492996621153306933527814537xy^{2}z^{13}+4728671266773594021761143967723370xz^{15}+y^{16}+580010544y^{14}z^{2}+231707043048132y^{12}z^{4}+6293018435817973752y^{10}z^{6}+43829965684618689177612y^{8}z^{8}+118016006426462224514796192y^{6}z^{10}+141641803611311180361836796666y^{4}z^{12}+74765824943967299766167023760352y^{2}z^{14}+13550260500909963959106243235607001z^{16}}{y^{2}(x^{2}y^{12}+108x^{2}y^{10}z^{2}-10206x^{2}y^{8}z^{4}-314928x^{2}y^{6}z^{6}+90876411x^{2}y^{4}z^{8}-401769396x^{2}y^{2}z^{10}+387420489x^{2}z^{12}+1134xy^{10}z^{3}+218700xy^{8}z^{5}+6259194xy^{6}z^{7}-580333572xy^{4}z^{9}+2453663097xy^{2}z^{11}-2324522934xz^{13}-144y^{12}z^{2}-27216y^{10}z^{4}-1161297y^{8}z^{6}-35429400y^{6}z^{8}-5299529652y^{4}z^{10}+24794911296y^{2}z^{12}-24407490807z^{14})}$

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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.bb.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.e.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.24.1.b.1 $48$ $2$ $2$ $1$ $0$ 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
48.96.1.i.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.z.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bj.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.by.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.do.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dy.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ec.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.eq.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.jj.2 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bfs.2 $48$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
240.96.1.oo.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ow.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.pu.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.qc.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.tm.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.tu.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.us.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.va.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.fd.2 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.ikt.2 $240$ $6$ $6$ $17$ $?$ not computed