Properties

Label 48.96.1-48.bk.2.9
Level $48$
Index $96$
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: $96$ $\PSL_2$-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: 48.96.1.1567

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}19&43\\40&17\end{bmatrix}$, $\begin{bmatrix}21&5\\4&1\end{bmatrix}$, $\begin{bmatrix}21&13\\28&21\end{bmatrix}$, $\begin{bmatrix}27&16\\4&1\end{bmatrix}$
Contains $-I$: no $\quad$ (see 48.48.1.bk.2 for the level structure with $-I$)
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $12288$

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 to other modular curves

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

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

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.48.1.bk.2 :

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

Equation of the image curve:

$0$ $=$ $ X^{4}+3X^{2}Y^{2}-9X^{2}Z^{2}+18Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
16.48.1-16.b.1.6 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0-24.by.1.6 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0-48.e.2.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0-48.e.2.26 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0-24.by.1.5 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1-16.b.1.10 $48$ $2$ $2$ $1$ $0$ 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
48.192.1-48.j.1.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.bc.1.5 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.bk.1.9 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.cb.1.5 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.cp.1.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.cs.1.5 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.de.1.9 $48$ $2$ $2$ $1$ $0$ dimension zero
48.192.1-48.dj.1.5 $48$ $2$ $2$ $1$ $0$ dimension zero
48.288.9-48.fa.2.1 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.384.9-48.bbb.2.18 $48$ $4$ $4$ $9$ $0$ $1^{4}\cdot2^{2}$
96.192.5-96.z.2.16 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5-96.bd.2.14 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5-96.bp.2.14 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5-96.cb.2.10 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5-96.cn.1.15 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5-96.cz.1.11 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5-96.dd.1.11 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5-96.dh.1.9 $96$ $2$ $2$ $5$ $?$ not computed
240.192.1-240.hx.2.3 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.ib.1.9 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.in.1.9 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.ir.1.9 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.jn.2.3 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.jv.1.9 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.kt.1.9 $240$ $2$ $2$ $1$ $?$ dimension zero
240.192.1-240.lb.1.9 $240$ $2$ $2$ $1$ $?$ dimension zero
240.480.17-240.cu.2.4 $240$ $5$ $5$ $17$ $?$ not computed