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Properties

Label 48.96.1.k.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: $32$
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: 48.96.1.1448

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&8\\32&35\end{bmatrix}$, $\begin{bmatrix}13&32\\32&21\end{bmatrix}$, $\begin{bmatrix}13&42\\24&13\end{bmatrix}$, $\begin{bmatrix}35&26\\0&29\end{bmatrix}$, $\begin{bmatrix}37&24\\8&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.k.2.1, 48.192.1-48.k.2.2, 48.192.1-48.k.2.3, 48.192.1-48.k.2.4, 48.192.1-48.k.2.5, 48.192.1-48.k.2.6, 48.192.1-48.k.2.7, 48.192.1-48.k.2.8, 48.192.1-48.k.2.9, 48.192.1-48.k.2.10, 48.192.1-48.k.2.11, 48.192.1-48.k.2.12, 48.192.1-48.k.2.13, 48.192.1-48.k.2.14, 48.192.1-48.k.2.15, 48.192.1-48.k.2.16, 240.192.1-48.k.2.1, 240.192.1-48.k.2.2, 240.192.1-48.k.2.3, 240.192.1-48.k.2.4, 240.192.1-48.k.2.5, 240.192.1-48.k.2.6, 240.192.1-48.k.2.7, 240.192.1-48.k.2.8, 240.192.1-48.k.2.9, 240.192.1-48.k.2.10, 240.192.1-48.k.2.11, 240.192.1-48.k.2.12, 240.192.1-48.k.2.13, 240.192.1-48.k.2.14, 240.192.1-48.k.2.15, 240.192.1-48.k.2.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

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

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

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

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{2^2}{3^2}\cdot\frac{1114512556032xy^{22}w-557256278016xy^{21}w^{2}+10618827964416xy^{20}w^{3}-5030785843200xy^{19}w^{4}+39526497386496xy^{18}w^{5}-17247855771648xy^{17}w^{6}+96163471392768xy^{16}w^{7}-39535204515840xy^{15}w^{8}+149121539923968xy^{14}w^{9}-55357733670912xy^{13}w^{10}+165783962179584xy^{12}w^{11}-56551038133248xy^{11}w^{12}+113473553204736xy^{10}w^{13}-31085462700288xy^{9}w^{14}+50268252288384xy^{8}w^{15}-11550885830208xy^{7}w^{16}+8489576446560xy^{6}w^{17}-581954223888xy^{5}w^{18}-5493099462696xy^{4}w^{19}+3720587640156xy^{3}w^{20}+12071994079686xy^{2}w^{21}-8690330203041xyw^{22}-185752092672y^{24}-4458050224128y^{22}w^{2}-185752092672y^{21}w^{3}-21846510010368y^{20}w^{4}-1491176521728y^{19}w^{5}-64991733350400y^{18}w^{6}-4227150053376y^{17}w^{7}-121253763317760y^{16}w^{8}-8718201372672y^{15}w^{9}-163432525160448y^{14}w^{10}-9103378378752y^{13}w^{11}-140360533972992y^{12}w^{12}-8298365303808y^{11}w^{13}-91957938992640y^{10}w^{14}-550956660480y^{9}w^{15}-19606575747456y^{8}w^{16}-1276582666176y^{7}w^{17}-13472981845344y^{6}w^{18}+1180251839376y^{5}w^{19}+5295051514728y^{4}w^{20}+1281969477828y^{3}w^{21}+4758367752842y^{2}w^{22}+1114512556032yz^{22}w+3343537668096yz^{21}w^{2}-3188744257536yz^{20}w^{3}-1408620036096yz^{19}w^{4}-11134805999616yz^{18}w^{5}-811375460352yz^{17}w^{6}+10916160307200yz^{16}w^{7}+10412651741184yz^{15}w^{8}+19225968648192yz^{14}w^{9}-28073227235328yz^{13}w^{10}+26010888708096yz^{12}w^{11}-33313443277824yz^{11}w^{12}+14425806398976yz^{10}w^{13}+12355189977600yz^{9}w^{14}-16192486051200yz^{8}w^{15}+78633748399680yz^{7}w^{16}-21733900608384yz^{6}w^{17}+89878185831312yz^{5}w^{18}-13849573142664yz^{4}w^{19}+45978350688936yz^{3}w^{20}-17095437996558yz^{2}w^{21}+4116033602925yzw^{22}+2896776734347yw^{23}-23914901687040z^{24}-96959871396864z^{22}w^{2}-6718034018304z^{21}w^{3}-217551083309568z^{20}w^{4}-6638057422848z^{19}w^{5}-315949549169664z^{18}w^{6}+54177693696z^{17}w^{7}-384797563282944z^{16}w^{8}-13621085577216z^{15}w^{9}-443074521443328z^{14}w^{10}+12434122598400z^{13}w^{11}-446180826539520z^{12}w^{12}+38982760151040z^{11}w^{13}-399898590554880z^{10}w^{14}+28820474199936z^{9}w^{15}-281770987388544z^{8}w^{16}+3683025278496z^{7}w^{17}-138060755867952z^{6}w^{18}-4656537492768z^{5}w^{19}-43095272462316z^{4}w^{20}-18153812950314z^{3}w^{21}-4392815476454z^{2}w^{22}-2896776734347zw^{23}-45349632w^{24}}{w^{4}(6449725440xy^{18}w-3224862720xy^{17}w^{2}+40633270272xy^{16}w^{3}-18704203776xy^{15}w^{4}+62562336768xy^{14}w^{5}-21929066496xy^{13}w^{6}+23160784896xy^{12}w^{7}-1063756800xy^{11}w^{8}-1139774976xy^{10}w^{9}-719684352xy^{9}w^{10}+1790481024xy^{8}w^{11}+33431616xy^{7}w^{12}-2053048032xy^{6}w^{13}+1569576528xy^{5}w^{14}+1832790024xy^{4}w^{15}-3889779948xy^{3}w^{16}-519451182xy^{2}w^{17}+6135070365xyw^{18}-1289945088y^{20}-20854112256y^{18}w^{2}-1074954240y^{17}w^{3}-55978242048y^{16}w^{4}-5159780352y^{15}w^{5}-43186286592y^{14}w^{6}-1970749440y^{13}w^{7}-11598308352y^{12}w^{8}+2386547712y^{11}w^{9}+196535808y^{10}w^{10}-2434344192y^{9}w^{11}-200010816y^{8}w^{12}+2948667840y^{7}w^{13}-663063840y^{6}w^{14}-2676440016y^{5}w^{15}+2075403672y^{4}w^{16}+713021004y^{3}w^{17}-3635492866y^{2}w^{18}-1289945088yz^{19}+3654844416yz^{17}w^{2}+21284093952yz^{16}w^{3}+7927787520yz^{15}w^{4}+55575134208yz^{14}w^{5}+34721021952yz^{13}w^{6}+76035096576yz^{12}w^{7}+51776216064yz^{11}w^{8}+87078136320yz^{10}w^{9}+42143556096yz^{9}w^{10}+76457063808yz^{8}w^{11}+26287400448yz^{7}w^{12}+43511513472yz^{6}w^{13}+10376248176yz^{5}w^{14}+15504882408yz^{4}w^{15}-1103961096yz^{3}w^{16}+2691402966yz^{2}w^{17}+2153276711yzw^{18}-2045023455yw^{19}-1289945088z^{20}+6449725440z^{19}w+17844240384z^{18}w^{2}+17199267840z^{17}w^{3}+32678608896z^{16}w^{4}+37945884672z^{15}w^{5}+48314714112z^{14}w^{6}+51324585984z^{13}w^{7}+68514149376z^{12}w^{8}+61029033984z^{11}w^{9}+71920141056z^{10}w^{10}+69721907328z^{9}w^{11}+59405996916z^{8}w^{12}+61644603744z^{7}w^{13}+40212786528z^{6}w^{14}+38196426528z^{5}w^{15}+16763569380z^{4}w^{16}+17205402690z^{3}w^{17}+1904321630z^{2}w^{18}+2045023455zw^{19})}$

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
16.48.1.b.2 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bb.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.d.2 $48$ $2$ $2$ $0$ $0$ full Jacobian

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.y.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.bg.3 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.bv.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.cb.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.cf.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.cl.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.cm.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.cn.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.288.17.ec.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.hz.2 $48$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
240.192.5.bes.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bet.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bew.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bex.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bfi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bfj.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bfm.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bfn.1 $240$ $2$ $2$ $5$ $?$ not computed