↑

Properties

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

Related objects

Downloads

Learn more

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: 16E1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.81

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}17&14\\8&25\end{bmatrix}$, $\begin{bmatrix}29&23\\36&5\end{bmatrix}$, $\begin{bmatrix}35&3\\32&5\end{bmatrix}$, $\begin{bmatrix}39&29\\28&7\end{bmatrix}$, $\begin{bmatrix}45&4\\40&45\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.f.1.1, 48.96.1-48.f.1.2, 48.96.1-48.f.1.3, 48.96.1-48.f.1.4, 48.96.1-48.f.1.5, 48.96.1-48.f.1.6, 48.96.1-48.f.1.7, 48.96.1-48.f.1.8, 48.96.1-48.f.1.9, 48.96.1-48.f.1.10, 48.96.1-48.f.1.11, 48.96.1-48.f.1.12, 48.96.1-48.f.1.13, 48.96.1-48.f.1.14, 48.96.1-48.f.1.15, 48.96.1-48.f.1.16, 240.96.1-48.f.1.1, 240.96.1-48.f.1.2, 240.96.1-48.f.1.3, 240.96.1-48.f.1.4, 240.96.1-48.f.1.5, 240.96.1-48.f.1.6, 240.96.1-48.f.1.7, 240.96.1-48.f.1.8, 240.96.1-48.f.1.9, 240.96.1-48.f.1.10, 240.96.1-48.f.1.11, 240.96.1-48.f.1.12, 240.96.1-48.f.1.13, 240.96.1-48.f.1.14, 240.96.1-48.f.1.15, 240.96.1-48.f.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 $ $=$ $ 6 x z + w^{2} $
$=$ $32 x^{2} - y^{2} - 2 z^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{4} + 2 x^{2} y^{2} - 4 z^{4} $
 Copy content Toggle raw display

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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^2}\cdot\frac{729y^{12}-54432y^{8}w^{4}+1458432y^{4}w^{8}+191056320z^{12}-253808640z^{8}w^{4}+112803840z^{4}w^{8}-16769024w^{12}}{w^{4}(9y^{8}+48y^{4}w^{4}-144z^{8}+64z^{4}w^{4})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.k.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.h.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.y.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.y.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.z.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.z.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ba.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ba.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bb.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bb.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.v.1 $48$ $3$ $3$ $9$ $3$ $1^{8}$
48.192.9.jb.1 $48$ $4$ $4$ $9$ $2$ $1^{8}$
240.96.1.dg.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.dg.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.dh.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.dh.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.di.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.di.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.dj.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.dj.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.l.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.xp.1 $240$ $6$ $6$ $17$ $?$ not computed