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Properties

Label 48.96.1.ea.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: $288$
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 $2^{2}\cdot4^{3}$
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: 48.96.1.874

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&45\\32&35\end{bmatrix}$, $\begin{bmatrix}9&11\\8&35\end{bmatrix}$, $\begin{bmatrix}13&33\\12&13\end{bmatrix}$, $\begin{bmatrix}19&47\\4&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.ea.1.1, 48.192.1-48.ea.1.2, 48.192.1-48.ea.1.3, 48.192.1-48.ea.1.4, 48.192.1-48.ea.1.5, 48.192.1-48.ea.1.6, 48.192.1-48.ea.1.7, 48.192.1-48.ea.1.8, 240.192.1-48.ea.1.1, 240.192.1-48.ea.1.2, 240.192.1-48.ea.1.3, 240.192.1-48.ea.1.4, 240.192.1-48.ea.1.5, 240.192.1-48.ea.1.6, 240.192.1-48.ea.1.7, 240.192.1-48.ea.1.8
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $12288$

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 $ $=$ $ x^{2} + 2 y z $
$=$ $14 x^{2} + 6 y^{2} - 8 y z + 6 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 12 x^{2} z^{2} - 6 y^{2} z^{2} + 4 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^4\cdot3^4}\cdot\frac{2422235760523644764160yz^{23}+1784061430721698332672yz^{21}w^{2}+487300855560381923328yz^{19}w^{4}+60395709292515164160yz^{17}w^{6}+3475874617250807808yz^{15}w^{8}+107790100775829504yz^{13}w^{10}+1944506844905472yz^{11}w^{12}+20851273580544yz^{9}w^{14}+129807442944yz^{7}w^{16}+434115072yz^{5}w^{18}+655488yz^{3}w^{20}+288yzw^{22}-415589953975640653824z^{24}-234730743802563133440z^{22}w^{2}-31415891817563947008z^{20}w^{4}+3725077708781125632z^{18}w^{6}+1111083893597601792z^{16}w^{8}+75389771733663744z^{14}w^{10}+2395719508230144z^{12}w^{12}+41611108368384z^{10}w^{14}+409786304256z^{8}w^{16}+2219215104z^{6}w^{18}+5937408z^{4}w^{20}+6048z^{2}w^{22}+w^{24}}{w^{2}z^{8}(1670162816581632yz^{13}+105321578379264yz^{11}w^{2}+2536708990464yz^{9}w^{4}+29066926464yz^{7}w^{6}+160000704yz^{5}w^{8}+371232yz^{3}w^{10}+240yzw^{12}-286554636582912z^{14}+31137317858304z^{12}w^{2}+2411546494464z^{10}w^{4}+56259288720z^{8}w^{6}+573268320z^{6}w^{8}+2559672z^{4}w^{10}+4056z^{2}w^{12}+w^{14})}$

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
16.48.0.bb.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bl.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.x.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.by.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.cb.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cc.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.ch.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
48.288.17.ckx.1 $48$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
48.384.17.blt.2 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$