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Properties

Label 48.96.1.b.1
Level $48$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $64$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{8}\cdot4^{4}\cdot16^{4}$ Cusp orbits $2^{4}\cdot8$
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: 16M1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.96.1.1462

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}11&38\\40&3\end{bmatrix}$, $\begin{bmatrix}21&44\\32&13\end{bmatrix}$, $\begin{bmatrix}25&10\\40&21\end{bmatrix}$, $\begin{bmatrix}37&26\\24&13\end{bmatrix}$, $\begin{bmatrix}43&8\\0&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.b.1.1, 48.192.1-48.b.1.2, 48.192.1-48.b.1.3, 48.192.1-48.b.1.4, 48.192.1-48.b.1.5, 48.192.1-48.b.1.6, 48.192.1-48.b.1.7, 48.192.1-48.b.1.8, 48.192.1-48.b.1.9, 48.192.1-48.b.1.10, 48.192.1-48.b.1.11, 48.192.1-48.b.1.12, 48.192.1-48.b.1.13, 48.192.1-48.b.1.14, 48.192.1-48.b.1.15, 48.192.1-48.b.1.16, 96.192.1-48.b.1.1, 96.192.1-48.b.1.2, 96.192.1-48.b.1.3, 96.192.1-48.b.1.4, 96.192.1-48.b.1.5, 96.192.1-48.b.1.6, 96.192.1-48.b.1.7, 96.192.1-48.b.1.8, 240.192.1-48.b.1.1, 240.192.1-48.b.1.2, 240.192.1-48.b.1.3, 240.192.1-48.b.1.4, 240.192.1-48.b.1.5, 240.192.1-48.b.1.6, 240.192.1-48.b.1.7, 240.192.1-48.b.1.8, 240.192.1-48.b.1.9, 240.192.1-48.b.1.10, 240.192.1-48.b.1.11, 240.192.1-48.b.1.12, 240.192.1-48.b.1.13, 240.192.1-48.b.1.14, 240.192.1-48.b.1.15, 240.192.1-48.b.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ 3 x y + z w $
$=$ $6 x^{2} - 3 y^{2} + z^{2} + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{4} - 9 x^{2} y^{2} - 2 x^{2} z^{2} - 6 y^{2} z^{2} $
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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:

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

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{12285y^{2}z^{22}+221130y^{2}z^{20}w^{2}+1521180y^{2}z^{18}w^{4}+5383800y^{2}z^{16}w^{6}+10571040y^{2}z^{14}w^{8}+8631360y^{2}z^{12}w^{10}-17262720y^{2}z^{10}w^{12}-84568320y^{2}z^{8}w^{14}-172281600y^{2}z^{6}w^{16}-194711040y^{2}z^{4}w^{18}-113218560y^{2}z^{2}w^{20}-25159680y^{2}w^{22}+z^{24}+8214z^{22}w^{2}+115596z^{20}w^{4}+633512z^{18}w^{6}+1874592z^{16}w^{8}+4499904z^{14}w^{10}+15715712z^{12}w^{12}+57697536z^{10}w^{14}+143799552z^{8}w^{16}+220302848z^{6}w^{18}+197139456z^{4}w^{20}+92276736z^{2}w^{22}+16777216w^{24}}{w^{4}z^{4}(z^{2}+2w^{2})^{2}(3y^{2}z^{10}+18y^{2}z^{8}w^{2}-24y^{2}z^{6}w^{4}+48y^{2}z^{4}w^{6}-144y^{2}z^{2}w^{8}-96y^{2}w^{10}-z^{12}-10z^{10}w^{2}-8z^{8}w^{4}+16z^{6}w^{6}-16z^{4}w^{8}-32z^{2}w^{10})}$

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
16.48.1.a.2 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bb.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.d.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
48.192.5.w.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.x.2 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.192.5.y.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ba.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.bg.4 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.bi.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.bq.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.br.1 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.288.17.dh.1 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.ho.2 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
96.192.5.c.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.c.3 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.d.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.d.3 $96$ $2$ $2$ $5$ $?$ not computed
96.192.9.o.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.o.4 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.p.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.p.4 $96$ $2$ $2$ $9$ $?$ not computed
240.192.5.bdh.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdj.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdk.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdl.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdm.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdn.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bdo.1 $240$ $2$ $2$ $5$ $?$ not computed