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Properties

Label 48.48.1.bx.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: 48.48.1.285

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}17&20\\28&11\end{bmatrix}$, $\begin{bmatrix}19&30\\28&17\end{bmatrix}$, $\begin{bmatrix}33&13\\16&15\end{bmatrix}$, $\begin{bmatrix}33&47\\28&21\end{bmatrix}$, $\begin{bmatrix}41&31\\12&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.bx.2.1, 48.96.1-48.bx.2.2, 48.96.1-48.bx.2.3, 48.96.1-48.bx.2.4, 48.96.1-48.bx.2.5, 48.96.1-48.bx.2.6, 48.96.1-48.bx.2.7, 48.96.1-48.bx.2.8, 48.96.1-48.bx.2.9, 48.96.1-48.bx.2.10, 48.96.1-48.bx.2.11, 48.96.1-48.bx.2.12, 48.96.1-48.bx.2.13, 48.96.1-48.bx.2.14, 48.96.1-48.bx.2.15, 48.96.1-48.bx.2.16, 240.96.1-48.bx.2.1, 240.96.1-48.bx.2.2, 240.96.1-48.bx.2.3, 240.96.1-48.bx.2.4, 240.96.1-48.bx.2.5, 240.96.1-48.bx.2.6, 240.96.1-48.bx.2.7, 240.96.1-48.bx.2.8, 240.96.1-48.bx.2.9, 240.96.1-48.bx.2.10, 240.96.1-48.bx.2.11, 240.96.1-48.bx.2.12, 240.96.1-48.bx.2.13, 240.96.1-48.bx.2.14, 240.96.1-48.bx.2.15, 240.96.1-48.bx.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
$(0:1:0)$, $(-6:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3}\cdot\frac{2088x^{2}y^{14}-3544571826x^{2}y^{12}z^{2}-6932242427688x^{2}y^{10}z^{4}+53011742336765199x^{2}y^{8}z^{6}+3003539166242014032x^{2}y^{6}z^{8}-276107538873630874545369x^{2}y^{4}z^{10}-318193573218156341529786828x^{2}y^{2}z^{12}-100516541203691949978889287705x^{2}z^{14}-1460844xy^{14}z-21002799096xy^{12}z^{3}+5451525458721xy^{10}z^{5}+772062233031073746xy^{8}z^{7}-567153163135642687704xy^{6}z^{9}-4110933851229296916667368xy^{4}z^{11}-3993284423003216365919256777xy^{2}z^{13}-1154460758489646977780788899690xz^{15}-y^{16}+347510736y^{14}z^{2}-841640347332y^{12}z^{4}+2668133519184168y^{10}z^{6}+4395738487801324788y^{8}z^{8}-16441666323193076500512y^{6}z^{10}-31815568820996942695406586y^{4}z^{12}-18253375230460820114149604352y^{2}z^{14}-3308169067604971667727148577241z^{16}}{y^{2}(x^{2}y^{12}+482652x^{2}y^{10}z^{2}+10805706018x^{2}y^{8}z^{4}+52918845228576x^{2}y^{6}z^{6}+82536342042418875x^{2}y^{4}z^{8}+38894063026481852508x^{2}y^{2}z^{10}+387420489x^{2}z^{12}+144xy^{12}z+17256078xy^{10}z^{3}+228725426700xy^{8}z^{5}+834645274821882xy^{6}z^{7}+1079026475131915020xy^{4}z^{9}+446709257654181508377xy^{2}z^{11}-2324522934xz^{13}+10224y^{12}z^{2}+456711696y^{10}z^{4}+3350448126639y^{8}z^{6}+7394410228074696y^{6}z^{8}+5727239278090175052y^{4}z^{10}+1280069276973550322256y^{2}z^{12}-24407490807z^{14})}$

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