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Properties

Label 48.192.1-48.du.2.2
Level $48$
Index $192$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
Index: $192$ $\PSL_2$-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: $1$
$\Q$-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: 48.192.1.743

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}9&20\\28&31\end{bmatrix}$, $\begin{bmatrix}27&25\\40&29\end{bmatrix}$, $\begin{bmatrix}39&31\\4&15\end{bmatrix}$
Contains $-I$: no $\quad$ (see 48.96.1.du.2 for the level structure with $-I$)
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $32$
Full 48-torsion field degree: $6144$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.c

Models

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

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

$ 0 $ $=$ $ x^{4} + 3 x^{3} z - 8 x^{2} y^{2} + 4 x^{2} z^{2} - 18 x y^{2} z + 3 x z^{3} + 4 y^{4} - 10 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 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{69060526080yz^{23}+2119892434944yz^{22}w+30759157186560yz^{21}w^{2}+280721908359168yz^{20}w^{3}+1808909791100928yz^{19}w^{4}+8757395261792256yz^{18}w^{5}+33091565654218752yz^{17}w^{6}+100088457614275584yz^{16}w^{7}+246534220793561088yz^{15}w^{8}+500487819241545216yz^{14}w^{9}+844216866764283648yz^{13}w^{10}+1189230024558246528yz^{12}w^{11}+1402562410853033472yz^{11}w^{12}+1385133682313934336yz^{10}w^{13}+1142925920080655136yz^{9}w^{14}+784205323514891376yz^{8}w^{15}+443954998217671872yz^{7}w^{16}+204964739131778496yz^{6}w^{17}+75870741223397856yz^{5}w^{18}+21963714849092880yz^{4}w^{19}+4787650041912960yz^{3}w^{20}+738677315452080yz^{2}w^{21}+71879937981072yzw^{22}+3315973726560yw^{23}-18504712192z^{24}-492189745152z^{23}w-5943991959552z^{22}w^{2}-42316394078208z^{21}w^{3}-188455085568000z^{20}w^{4}-463856032481280z^{19}w^{5}+120261155840000z^{18}w^{6}+6659181227910144z^{17}w^{7}+33743552777608704z^{16}w^{8}+108157328297335296z^{15}w^{9}+258418750477146624z^{14}w^{10}+486792047520847488z^{13}w^{11}+742064977594221760z^{12}w^{12}+927800861087061120z^{11}w^{13}+957631924929761280z^{10}w^{14}+817400011087197744z^{9}w^{15}+575740380099649248z^{8}w^{16}+332637383034052992z^{7}w^{17}+156013476439466272z^{6}w^{18}+58457001088039248z^{5}w^{19}+17079956528983584z^{4}w^{20}+3748875036979440z^{3}w^{21}+581295670033680z^{2}w^{22}+56758217064960zw^{23}+2623889712383w^{24}}{(z+w)^{8}(101162880yz^{15}+1996712896yz^{14}w+18195006224yz^{13}w^{2}+101535634040yz^{12}w^{3}+388034594048yz^{11}w^{4}+1075726769176yz^{10}w^{5}+2234806668750yz^{9}w^{6}+3542962956845yz^{8}w^{7}+4321795117864yz^{7}w^{8}+4056619966944yz^{6}w^{9}+2906346856180yz^{5}w^{10}+1560960032786yz^{4}w^{11}+608453574664yz^{3}w^{12}+162522270444yz^{2}w^{13}+26603501880yzw^{14}+2012139360yw^{15}-27106512z^{16}-423933056z^{15}w-2726599104z^{14}w^{2}-8025782472z^{13}w^{3}+834506068z^{12}w^{4}+103822273392z^{11}w^{5}+464736234632z^{10}w^{6}+1212153600217z^{9}w^{7}+2193514528149z^{8}w^{8}+2910984673280z^{7}w^{9}+2890602689772z^{6}w^{10}+2152167661170z^{5}w^{11}+1186815145784z^{4}w^{12}+470951351268z^{3}w^{13}+127266775860z^{2}w^{14}+20978512272zw^{15}+1592181423w^{16})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.1.du.2 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}x$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ X^{4}-8X^{2}Y^{2}+4Y^{4}+3X^{3}Z-18XY^{2}Z+4X^{2}Z^{2}-10Y^{2}Z^{2}+3XZ^{3}+Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
16.96.0-16.z.1.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-16.z.1.7 $48$ $2$ $2$ $0$ $0$ full Jacobian
24.96.0-24.bj.2.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-24.bj.2.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-48.u.1.4 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-48.u.1.5 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-48.bz.1.3 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-48.bz.1.10 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.1-48.bp.1.5 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1-48.bp.1.10 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1-48.bt.2.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1-48.bt.2.11 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1-48.cg.1.3 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1-48.cg.1.8 $48$ $2$ $2$ $1$ $1$ 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
48.576.17-48.ckj.2.4 $48$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
48.768.17-48.blg.1.2 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$