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Properties

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

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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: $8$ (none of which are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.63

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&23\\12&13\end{bmatrix}$, $\begin{bmatrix}23&11\\0&29\end{bmatrix}$, $\begin{bmatrix}23&44\\0&47\end{bmatrix}$, $\begin{bmatrix}35&34\\24&7\end{bmatrix}$, $\begin{bmatrix}35&39\\20&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.ca.1.1, 48.96.1-48.ca.1.2, 48.96.1-48.ca.1.3, 48.96.1-48.ca.1.4, 48.96.1-48.ca.1.5, 48.96.1-48.ca.1.6, 48.96.1-48.ca.1.7, 48.96.1-48.ca.1.8, 48.96.1-48.ca.1.9, 48.96.1-48.ca.1.10, 48.96.1-48.ca.1.11, 48.96.1-48.ca.1.12, 48.96.1-48.ca.1.13, 48.96.1-48.ca.1.14, 48.96.1-48.ca.1.15, 48.96.1-48.ca.1.16, 240.96.1-48.ca.1.1, 240.96.1-48.ca.1.2, 240.96.1-48.ca.1.3, 240.96.1-48.ca.1.4, 240.96.1-48.ca.1.5, 240.96.1-48.ca.1.6, 240.96.1-48.ca.1.7, 240.96.1-48.ca.1.8, 240.96.1-48.ca.1.9, 240.96.1-48.ca.1.10, 240.96.1-48.ca.1.11, 240.96.1-48.ca.1.12, 240.96.1-48.ca.1.13, 240.96.1-48.ca.1.14, 240.96.1-48.ca.1.15, 240.96.1-48.ca.1.16
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

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 4 x y + y^{2} + w^{2} $
$=$ $12 x^{2} + x y + y^{2} - z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 2 x^{4} - 3 x^{2} y^{2} + 3 x^{2} z^{2} + z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{3}z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^2}\cdot\frac{193536y^{2}z^{10}-20736y^{2}z^{8}w^{2}-24541056y^{2}z^{6}w^{4}+325448928y^{2}z^{4}w^{6}-764583948y^{2}z^{2}w^{8}+191102247y^{2}w^{10}-131072z^{12}+589824z^{10}w^{2}+2730240z^{8}w^{4}-5177088z^{6}w^{6}-14059008z^{4}w^{8}-254975040z^{2}w^{10}+95550759w^{12}}{w^{2}z^{2}(64y^{2}z^{6}-528y^{2}z^{4}w^{2}+252y^{2}z^{2}w^{4}-27y^{2}w^{6}+512z^{6}w^{2}-816z^{4}w^{4}+288z^{2}w^{6}-27w^{8})}$

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.ba.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.f.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.g.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ba.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bn.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bx.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dl.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dz.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ed.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.en.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.js.2 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bft.1 $48$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
240.96.1.op.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ox.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.pv.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.qd.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.tn.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.tv.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ut.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.vb.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.fm.2 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.ilk.1 $240$ $6$ $6$ $17$ $?$ not computed