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Properties

Label 48.48.1.bm.2
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: $32$
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: 48.48.1.432

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}11&42\\36&37\end{bmatrix}$, $\begin{bmatrix}11&46\\40&43\end{bmatrix}$, $\begin{bmatrix}17&7\\24&23\end{bmatrix}$, $\begin{bmatrix}41&42\\8&1\end{bmatrix}$, $\begin{bmatrix}47&12\\36&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.bm.2.1, 48.96.1-48.bm.2.2, 48.96.1-48.bm.2.3, 48.96.1-48.bm.2.4, 48.96.1-48.bm.2.5, 48.96.1-48.bm.2.6, 48.96.1-48.bm.2.7, 48.96.1-48.bm.2.8, 48.96.1-48.bm.2.9, 48.96.1-48.bm.2.10, 48.96.1-48.bm.2.11, 48.96.1-48.bm.2.12, 48.96.1-48.bm.2.13, 48.96.1-48.bm.2.14, 48.96.1-48.bm.2.15, 48.96.1-48.bm.2.16, 96.96.1-48.bm.2.1, 96.96.1-48.bm.2.2, 96.96.1-48.bm.2.3, 96.96.1-48.bm.2.4, 96.96.1-48.bm.2.5, 96.96.1-48.bm.2.6, 96.96.1-48.bm.2.7, 96.96.1-48.bm.2.8, 240.96.1-48.bm.2.1, 240.96.1-48.bm.2.2, 240.96.1-48.bm.2.3, 240.96.1-48.bm.2.4, 240.96.1-48.bm.2.5, 240.96.1-48.bm.2.6, 240.96.1-48.bm.2.7, 240.96.1-48.bm.2.8, 240.96.1-48.bm.2.9, 240.96.1-48.bm.2.10, 240.96.1-48.bm.2.11, 240.96.1-48.bm.2.12, 240.96.1-48.bm.2.13, 240.96.1-48.bm.2.14, 240.96.1-48.bm.2.15, 240.96.1-48.bm.2.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

$ 0 $ $=$ $ 24 x y - 6 y^{2} + w^{2} $
$=$ $24 x^{2} + 6 x y - z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 3 x^{2} y^{2} - 9 x^{2} z^{2} + 18 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{6}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{387072y^{2}z^{10}+13824y^{2}z^{8}w^{2}-5453568y^{2}z^{6}w^{4}-24107328y^{2}z^{4}w^{6}-18878616y^{2}z^{2}w^{8}-1572858y^{2}w^{10}-131072z^{12}-196608z^{10}w^{2}+303360z^{8}w^{4}+191744z^{6}w^{6}-173568z^{4}w^{8}+1049280z^{2}w^{10}+131071w^{12}}{w^{2}z^{2}(384y^{2}z^{6}+1056y^{2}z^{4}w^{2}+168y^{2}z^{2}w^{4}+6y^{2}w^{6}-512z^{6}w^{2}-272z^{4}w^{4}-32z^{2}w^{6}-w^{8})}$

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
16.24.1.b.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.24.0.by.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.f.1 $48$ $2$ $2$ $0$ $0$ full Jacobian

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.j.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bd.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bo.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cb.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cn.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.cu.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dg.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dh.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.fk.1 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bbd.2 $48$ $4$ $4$ $9$ $0$ $1^{4}\cdot2^{2}$
96.96.5.bb.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.bf.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.br.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.cd.1 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.cp.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.db.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.df.2 $96$ $2$ $2$ $5$ $?$ not computed
96.96.5.dj.2 $96$ $2$ $2$ $5$ $?$ not computed
240.96.1.hz.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.id.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ip.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.it.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.jp.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.jx.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.kv.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ld.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.da.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.eho.2 $240$ $6$ $6$ $17$ $?$ not computed