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Properties

Label 48.96.1.dy.2
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 $2^{4}\cdot4^{2}$
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: 48.96.1.952

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&8\\0&1\end{bmatrix}$, $\begin{bmatrix}11&0\\32&7\end{bmatrix}$, $\begin{bmatrix}23&34\\44&45\end{bmatrix}$, $\begin{bmatrix}25&33\\28&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.dy.2.1, 48.192.1-48.dy.2.2, 48.192.1-48.dy.2.3, 48.192.1-48.dy.2.4, 48.192.1-48.dy.2.5, 48.192.1-48.dy.2.6, 48.192.1-48.dy.2.7, 48.192.1-48.dy.2.8, 240.192.1-48.dy.2.1, 240.192.1-48.dy.2.2, 240.192.1-48.dy.2.3, 240.192.1-48.dy.2.4, 240.192.1-48.dy.2.5, 240.192.1-48.dy.2.6, 240.192.1-48.dy.2.7, 240.192.1-48.dy.2.8
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 $ $=$ $ 2 x^{2} - y^{2} + 3 y z - z^{2} + w^{2} $
$=$ $8 x^{2} + y^{2} - 2 y z + z^{2} - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 28 x^{2} y^{2} - 4 x^{2} z^{2} + 100 y^{4} - 20 y^{2} z^{2} + 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle x$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{5^2}\cdot\frac{822206626164yz^{23}+80797745240340yz^{21}w^{2}+2806931728903500yz^{19}w^{4}+39473588445142500yz^{17}w^{6}+187291996449690000yz^{15}w^{8}+431163970297350000yz^{13}w^{10}+548697785458500000yz^{11}w^{12}+400436040262500000yz^{9}w^{14}+163913542968750000yz^{7}w^{16}+34563855093750000yz^{5}w^{18}+3095331093750000yz^{3}w^{20}+73150781250000yzw^{22}-345494596255z^{24}-33448046602644z^{22}w^{2}-1130163120416970z^{20}w^{4}-14883430762869500z^{18}w^{6}-55055791364063250z^{16}w^{8}-73709655231150000z^{14}w^{10}+2092704574150000z^{12}w^{12}+104168428692000000z^{10}w^{14}+108422913782812500z^{8}w^{16}+47052787081250000z^{6}w^{18}+8780600859375000z^{4}w^{20}+549293906250000z^{2}w^{22}+4752734375000w^{24}}{w^{2}(1105582708016yz^{21}+231692731000yz^{19}w^{2}-1582777407600yz^{17}w^{4}+510691727000yz^{15}w^{6}+564080675000yz^{13}w^{8}-475984725000yz^{11}w^{10}+119925625000yz^{9}w^{12}+9893125000yz^{7}w^{14}-13003125000yz^{5}w^{16}+2968750000yz^{3}w^{18}-234375000yzw^{20}-464570378720z^{22}+579669831564z^{20}w^{2}+595403983480z^{18}w^{4}-1095971907575z^{16}w^{6}+303071565000z^{14}w^{8}+327062112500z^{12}w^{10}-306860275000z^{10}w^{12}+117067218750z^{8}w^{14}-25923125000z^{6}w^{16}+4953125000z^{4}w^{18}-937500000z^{2}w^{20}+87890625w^{22})}$

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.z.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bl.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.w.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bw.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bw.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bz.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.ckr.1 $48$ $3$ $3$ $17$ $3$ $1^{8}\cdot2^{4}$
48.384.17.blp.1 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$