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Properties

Label 48.96.1.bg.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 $2^{6}\cdot4$
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: 48.96.1.730

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}7&15\\16&29\end{bmatrix}$, $\begin{bmatrix}11&21\\8&29\end{bmatrix}$, $\begin{bmatrix}13&10\\8&21\end{bmatrix}$, $\begin{bmatrix}29&31\\40&39\end{bmatrix}$, $\begin{bmatrix}37&28\\24&25\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.bg.1.1, 48.192.1-48.bg.1.2, 48.192.1-48.bg.1.3, 48.192.1-48.bg.1.4, 48.192.1-48.bg.1.5, 48.192.1-48.bg.1.6, 48.192.1-48.bg.1.7, 48.192.1-48.bg.1.8, 48.192.1-48.bg.1.9, 48.192.1-48.bg.1.10, 48.192.1-48.bg.1.11, 48.192.1-48.bg.1.12, 96.192.1-48.bg.1.1, 96.192.1-48.bg.1.2, 96.192.1-48.bg.1.3, 96.192.1-48.bg.1.4, 96.192.1-48.bg.1.5, 96.192.1-48.bg.1.6, 96.192.1-48.bg.1.7, 96.192.1-48.bg.1.8, 240.192.1-48.bg.1.1, 240.192.1-48.bg.1.2, 240.192.1-48.bg.1.3, 240.192.1-48.bg.1.4, 240.192.1-48.bg.1.5, 240.192.1-48.bg.1.6, 240.192.1-48.bg.1.7, 240.192.1-48.bg.1.8, 240.192.1-48.bg.1.9, 240.192.1-48.bg.1.10, 240.192.1-48.bg.1.11, 240.192.1-48.bg.1.12
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 y + x w + y z $
$=$ $6 x^{2} - y^{2} - 2 y w - 3 z^{2} + w^{2}$
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Singular plane model Singular plane model

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

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

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{1}{3}\cdot\frac{14731544520xz^{23}-45460171728xz^{21}w^{2}+29558512224xz^{19}w^{4}-40831045056xz^{17}w^{6}+146040931584xz^{15}w^{8}+31730504855040xz^{13}w^{10}+8808022010880xz^{11}w^{12}-7936962951168xz^{9}w^{14}+1018291709952xz^{7}w^{16}+25676476416xz^{5}w^{18}-4927463424xz^{3}w^{20}-170311680xzw^{22}+1438433640y^{2}z^{22}-4268534112y^{2}z^{20}w^{2}-1080203040y^{2}z^{18}w^{4}+208619224704y^{2}z^{16}w^{6}-3258023378688y^{2}z^{14}w^{8}-18165704702976y^{2}z^{12}w^{10}+2861161159680y^{2}z^{10}w^{12}+1242821283840y^{2}z^{8}w^{14}-238599862272y^{2}z^{6}w^{16}-2271928320y^{2}z^{4}w^{18}+1307271168y^{2}z^{2}w^{20}+40140800y^{2}w^{22}+3711583944yz^{22}w-9926845488yz^{20}w^{3}-33023979936yz^{18}w^{5}-121634851392yz^{16}w^{7}+20714654299392yz^{14}w^{9}+4339495088640yz^{12}w^{11}-7872445043712yz^{10}w^{13}+1946625177600yz^{8}w^{15}-116297275392yz^{6}w^{17}-7171338240yz^{4}w^{19}+558686208yz^{2}w^{21}+23511040yw^{23}+10416775041z^{24}-32724011016z^{22}w^{2}+20410641144z^{20}w^{4}+71257498848z^{18}w^{6}-279454195152z^{16}w^{8}-13239129141504z^{14}w^{10}+3352890703104z^{12}w^{12}+2521791138816z^{10}w^{14}-1010918009088z^{8}w^{16}+91286673408z^{6}w^{18}+3449862144z^{4}w^{20}-455712768z^{2}w^{22}-16625664w^{24}}{w^{2}z^{2}(14722884xz^{17}w^{2}-108965088xz^{15}w^{4}-666247680xz^{13}w^{6}-2600325504xz^{11}w^{8}-3608841600xz^{9}w^{10}-2102340096xz^{7}w^{12}-469628928xz^{5}w^{14}-31721472xz^{3}w^{16}-3072xzw^{18}+19683y^{2}z^{18}+5471874y^{2}z^{16}w^{2}+88494768y^{2}z^{14}w^{4}+631232352y^{2}z^{12}w^{6}+1592641440y^{2}z^{10}w^{8}+1696054464y^{2}z^{8}w^{10}+764944128y^{2}z^{6}w^{12}+137000448y^{2}z^{4}w^{14}+7502592y^{2}z^{2}w^{16}+512y^{2}w^{18}-708588yz^{18}w-34720812yz^{16}w^{3}-452446560yz^{14}w^{5}-1615230720yz^{12}w^{7}-2206953216yz^{10}w^{9}-1023684480yz^{8}w^{11}-56498688yz^{6}w^{13}+42448896yz^{4}w^{15}+4365312yz^{2}w^{17}+1024yw^{19}+12006630z^{18}w^{2}-223074z^{16}w^{4}+333193824z^{14}w^{6}+902630304z^{12}w^{8}+1025234496z^{10}w^{10}+273575232z^{8}w^{12}-91556352z^{6}w^{14}-41080320z^{4}w^{16}-3095040z^{2}w^{18}-512w^{20})}$

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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.j.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.be.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bc.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bd.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.h.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bw.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bx.2 $48$ $2$ $2$ $1$ $0$ 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
48.192.5.gm.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.go.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.gp.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.gs.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.288.17.mc.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.ou.2 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
96.192.5.bb.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.be.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.bl.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.bm.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.cl.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.cm.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ct.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.cw.1 $96$ $2$ $2$ $5$ $?$ not computed
240.192.5.brb.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.brc.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.brh.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.brj.2 $240$ $2$ $2$ $5$ $?$ not computed