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Properties

Label 48.48.1.bx.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: 16G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.239

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&16\\44&23\end{bmatrix}$, $\begin{bmatrix}5&23\\12&37\end{bmatrix}$, $\begin{bmatrix}7&5\\40&21\end{bmatrix}$, $\begin{bmatrix}13&6\\16&25\end{bmatrix}$, $\begin{bmatrix}41&31\\20&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.bx.1.1, 48.96.1-48.bx.1.2, 48.96.1-48.bx.1.3, 48.96.1-48.bx.1.4, 48.96.1-48.bx.1.5, 48.96.1-48.bx.1.6, 48.96.1-48.bx.1.7, 48.96.1-48.bx.1.8, 48.96.1-48.bx.1.9, 48.96.1-48.bx.1.10, 48.96.1-48.bx.1.11, 48.96.1-48.bx.1.12, 48.96.1-48.bx.1.13, 48.96.1-48.bx.1.14, 48.96.1-48.bx.1.15, 48.96.1-48.bx.1.16, 240.96.1-48.bx.1.1, 240.96.1-48.bx.1.2, 240.96.1-48.bx.1.3, 240.96.1-48.bx.1.4, 240.96.1-48.bx.1.5, 240.96.1-48.bx.1.6, 240.96.1-48.bx.1.7, 240.96.1-48.bx.1.8, 240.96.1-48.bx.1.9, 240.96.1-48.bx.1.10, 240.96.1-48.bx.1.11, 240.96.1-48.bx.1.12, 240.96.1-48.bx.1.13, 240.96.1-48.bx.1.14, 240.96.1-48.bx.1.15, 240.96.1-48.bx.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
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
$(0:1:0)$, $(-6:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^2}\cdot\frac{72x^{2}y^{14}-1867374x^{2}y^{12}z^{2}+26731141128x^{2}y^{10}z^{4}-6855185486148399x^{2}y^{8}z^{6}-6972702061285216512x^{2}y^{6}z^{8}+3603467780380039981689x^{2}y^{4}z^{10}-524312015331862950997572x^{2}y^{2}z^{12}+24540171192307636348712985x^{2}z^{14}-3636xy^{14}z+562222296xy^{12}z^{3}-10044857414241xy^{10}z^{5}-107473302283556466xy^{8}z^{7}-67226141300667384456xy^{6}z^{9}+39587542997756743507608xy^{4}z^{11}-5939170412867590801673943xy^{2}z^{13}+281850771115636280944373610xz^{15}+y^{16}-390096y^{14}z^{2}+17555652612y^{12}z^{4}-350621539089048y^{10}z^{6}-840046440149555508y^{8}z^{8}+59683491801367516512y^{6}z^{10}+77544013593479198063706y^{4}z^{12}-15356314015030356818143248y^{2}z^{14}+807658463770743059542110681z^{16}}{zy^{2}(3015x^{2}y^{10}z+49311504x^{2}y^{8}z^{3}+221548784013x^{2}y^{6}z^{5}+404057240293356x^{2}y^{4}z^{7}+321578631812348793x^{2}y^{2}z^{9}+92900616229677957120x^{2}z^{11}+xy^{12}+95634xy^{10}z^{2}+1000196748xy^{8}z^{4}+3566049101712xy^{6}z^{6}+5601923216587881xy^{4}z^{8}+4006497033521732394xy^{2}z^{10}+1066989717238031843328xz^{12}+72y^{12}z+2246454y^{10}z^{3}+14046103728y^{8}z^{5}+33582383063694y^{6}z^{7}+36432858149389728y^{4}z^{9}+17775226735718178105y^{2}z^{11}+3057516119159784603648z^{13})}$

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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
16.24.0.e.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.bz.1 $24$ $2$ $2$ $0$ $0$ 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.r.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.y.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bg.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ca.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dm.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dx.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.eb.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.eo.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.jh.2 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bfq.1 $48$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
240.96.1.om.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ou.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ps.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.qa.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.tk.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ts.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.uq.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.uy.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.fb.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.ikr.1 $240$ $6$ $6$ $17$ $?$ not computed