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Properties

Label 48.48.1.h.1
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: 16E1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.30

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&13\\16&29\end{bmatrix}$, $\begin{bmatrix}5&18\\24&5\end{bmatrix}$, $\begin{bmatrix}17&14\\16&21\end{bmatrix}$, $\begin{bmatrix}19&9\\16&25\end{bmatrix}$, $\begin{bmatrix}35&16\\24&43\end{bmatrix}$, $\begin{bmatrix}45&37\\40&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.h.1.1, 48.96.1-48.h.1.2, 48.96.1-48.h.1.3, 48.96.1-48.h.1.4, 48.96.1-48.h.1.5, 48.96.1-48.h.1.6, 48.96.1-48.h.1.7, 48.96.1-48.h.1.8, 48.96.1-48.h.1.9, 48.96.1-48.h.1.10, 48.96.1-48.h.1.11, 48.96.1-48.h.1.12, 48.96.1-48.h.1.13, 48.96.1-48.h.1.14, 48.96.1-48.h.1.15, 48.96.1-48.h.1.16, 48.96.1-48.h.1.17, 48.96.1-48.h.1.18, 48.96.1-48.h.1.19, 48.96.1-48.h.1.20, 48.96.1-48.h.1.21, 48.96.1-48.h.1.22, 48.96.1-48.h.1.23, 48.96.1-48.h.1.24, 240.96.1-48.h.1.1, 240.96.1-48.h.1.2, 240.96.1-48.h.1.3, 240.96.1-48.h.1.4, 240.96.1-48.h.1.5, 240.96.1-48.h.1.6, 240.96.1-48.h.1.7, 240.96.1-48.h.1.8, 240.96.1-48.h.1.9, 240.96.1-48.h.1.10, 240.96.1-48.h.1.11, 240.96.1-48.h.1.12, 240.96.1-48.h.1.13, 240.96.1-48.h.1.14, 240.96.1-48.h.1.15, 240.96.1-48.h.1.16, 240.96.1-48.h.1.17, 240.96.1-48.h.1.18, 240.96.1-48.h.1.19, 240.96.1-48.h.1.20, 240.96.1-48.h.1.21, 240.96.1-48.h.1.22, 240.96.1-48.h.1.23, 240.96.1-48.h.1.24
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} + 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
$(0:1:0)$, $(0:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^2}\cdot\frac{16187040x^{2}y^{12}z^{2}-962206179757056x^{2}y^{8}z^{6}+319479483107509862400x^{2}y^{4}z^{10}-2208245624028041828106240x^{2}z^{14}-6768xy^{14}z+5634505338624xy^{10}z^{5}-8217092592118923264xy^{6}z^{9}+306700803051956449837056xy^{2}z^{13}+y^{16}-15651595008y^{12}z^{4}+91326841956974592y^{8}z^{8}-6815591803818736091136y^{4}z^{12}+4738381338321616896z^{16}}{zy^{4}(36x^{2}y^{8}z-1679616x^{2}y^{4}z^{5}-78364164096x^{2}z^{9}-xy^{10}-2176782336xy^{2}z^{8}+2592y^{8}z^{3}-120932352y^{4}z^{7}-2821109907456z^{11})}$

Modular covers

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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.q.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.g.1 $48$ $2$ $2$ $0$ $0$ 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.bg.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bg.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bj.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bj.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bn.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bn.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bq.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bq.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.3.fp.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.96.3.fs.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.fs.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.fv.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.fw.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.gb.1 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.gb.2 $48$ $2$ $2$ $3$ $0$ $2$
48.96.3.gd.1 $48$ $2$ $2$ $3$ $1$ $1^{2}$
48.144.9.be.1 $48$ $3$ $3$ $9$ $1$ $1^{8}$
48.192.9.mk.1 $48$ $4$ $4$ $9$ $1$ $1^{8}$
240.96.1.dw.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.dw.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ed.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ed.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.el.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.el.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.eq.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.eq.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.3.rw.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ry.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.ry.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.rz.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.si.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.sm.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.sm.2 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.so.1 $240$ $2$ $2$ $3$ $?$ not computed
240.240.17.p.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.bfr.1 $240$ $6$ $6$ $17$ $?$ not computed