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Properties

Label 48.96.1.cd.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: $12$ Newform level: $24$
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 $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.96.1.2628

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}17&41\\30&41\end{bmatrix}$, $\begin{bmatrix}25&36\\24&41\end{bmatrix}$, $\begin{bmatrix}29&3\\30&19\end{bmatrix}$, $\begin{bmatrix}35&22\\36&37\end{bmatrix}$, $\begin{bmatrix}43&15\\30&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.cd.2.1, 48.192.1-48.cd.2.2, 48.192.1-48.cd.2.3, 48.192.1-48.cd.2.4, 48.192.1-48.cd.2.5, 48.192.1-48.cd.2.6, 48.192.1-48.cd.2.7, 48.192.1-48.cd.2.8, 48.192.1-48.cd.2.9, 48.192.1-48.cd.2.10, 48.192.1-48.cd.2.11, 48.192.1-48.cd.2.12, 48.192.1-48.cd.2.13, 48.192.1-48.cd.2.14, 48.192.1-48.cd.2.15, 48.192.1-48.cd.2.16, 240.192.1-48.cd.2.1, 240.192.1-48.cd.2.2, 240.192.1-48.cd.2.3, 240.192.1-48.cd.2.4, 240.192.1-48.cd.2.5, 240.192.1-48.cd.2.6, 240.192.1-48.cd.2.7, 240.192.1-48.cd.2.8, 240.192.1-48.cd.2.9, 240.192.1-48.cd.2.10, 240.192.1-48.cd.2.11, 240.192.1-48.cd.2.12, 240.192.1-48.cd.2.13, 240.192.1-48.cd.2.14, 240.192.1-48.cd.2.15, 240.192.1-48.cd.2.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

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

$ 0 $ $=$ $ 3 x^{2} - 2 y^{2} + z^{2} + w^{2} $
$=$ $4 x y - z^{2} + 2 z w + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{4} - 12 x^{3} z + 4 x^{2} y^{2} + 6 x^{2} z^{2} + 12 x z^{3} - 2 y^{4} + 4 y^{2} z^{2} + 3 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 2y$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^4\,\frac{(5z^{4}-12z^{3}w+10z^{2}w^{2}+12zw^{3}+5w^{4})(2451232y^{2}z^{18}-31339008y^{2}z^{17}w+163978272y^{2}z^{16}w^{2}-451464192y^{2}z^{15}w^{3}+663295104y^{2}z^{14}w^{4}-438746112y^{2}z^{13}w^{5}-100076928y^{2}z^{12}w^{6}+351544320y^{2}z^{11}w^{7}-102132288y^{2}z^{10}w^{8}-102132288y^{2}z^{8}w^{10}-351544320y^{2}z^{7}w^{11}-100076928y^{2}z^{6}w^{12}+438746112y^{2}z^{5}w^{13}+663295104y^{2}z^{4}w^{14}+451464192y^{2}z^{3}w^{15}+163978272y^{2}z^{2}w^{16}+31339008y^{2}zw^{17}+2451232y^{2}w^{18}-1591433z^{20}+21392652z^{19}w-121862906z^{18}w^{2}+379092180z^{17}w^{3}-687113109z^{16}w^{4}+688300656z^{15}w^{5}-231357624z^{14}w^{6}-244556592z^{13}w^{7}+281297694z^{12}w^{8}-77493720z^{11}w^{9}-108372636z^{10}w^{10}+77493720z^{9}w^{11}+281297694z^{8}w^{12}+244556592z^{7}w^{13}-231357624z^{6}w^{14}-688300656z^{5}w^{15}-687113109z^{4}w^{16}-379092180z^{3}w^{17}-121862906z^{2}w^{18}-21392652zw^{19}-1591433w^{20})}{(z^{2}-2zw-w^{2})^{2}(z^{2}+2zw-w^{2})^{4}(560y^{2}z^{10}-4224y^{2}z^{9}w+9712y^{2}z^{8}w^{2}-8448y^{2}z^{7}w^{3}-1312y^{2}z^{6}w^{4}-1312y^{2}z^{4}w^{6}+8448y^{2}z^{3}w^{7}+9712y^{2}z^{2}w^{8}+4224y^{2}zw^{9}+560y^{2}w^{10}+83z^{12}-852z^{11}w+3522z^{10}w^{2}-4284z^{9}w^{3}+1245z^{8}w^{4}+3480z^{7}w^{5}-4388z^{6}w^{6}-3480z^{5}w^{7}+1245z^{4}w^{8}+4284z^{3}w^{9}+3522z^{2}w^{10}+852zw^{11}+83w^{12})}$

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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
24.48.1.dr.1 $24$ $2$ $2$ $1$ $0$ 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
48.192.9.on.2 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
48.192.9.ow.4 $48$ $2$ $2$ $9$ $3$ $1^{4}\cdot4$
48.192.9.ox.4 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
48.192.9.oy.2 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
48.192.9.pm.3 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
48.192.9.pv.1 $48$ $2$ $2$ $9$ $3$ $1^{4}\cdot4$
48.192.9.pw.1 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
48.192.9.px.3 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
48.288.9.a.2 $48$ $3$ $3$ $9$ $2$ $1^{4}\cdot4$
144.288.9.a.4 $144$ $3$ $3$ $9$ $?$ not computed
144.288.17.d.3 $144$ $3$ $3$ $17$ $?$ not computed
144.288.17.h.2 $144$ $3$ $3$ $17$ $?$ not computed
240.192.9.cmn.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cmo.4 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cmp.4 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cmq.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.cog.3 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.coh.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.coi.1 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.coj.3 $240$ $2$ $2$ $9$ $?$ not computed