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Properties

Label 48.192.3-48.cj.1.1
Level $48$
Index $192$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
Index: $192$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $4^{8}\cdot16^{4}$ Cusp orbits $2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16J3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.192.3.32

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&34\\0&37\end{bmatrix}$, $\begin{bmatrix}11&46\\0&17\end{bmatrix}$, $\begin{bmatrix}19&44\\0&17\end{bmatrix}$, $\begin{bmatrix}33&20\\8&1\end{bmatrix}$
Contains $-I$: no $\quad$ (see 48.96.3.cj.1 for the level structure with $-I$)
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $6144$

Jacobian

Conductor: $2^{17}\cdot3^{6}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 288.2.a.d, 576.2.d.a

Models

Embedded model Embedded model in $\mathbb{P}^{5}$

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

$ 0 $ $=$ $ 3 x^{8} + 10 x^{7} y - 2 x^{7} z + 13 x^{6} y^{2} - 4 x^{6} y z + 28 x^{6} z^{2} + 8 x^{5} y^{3} + \cdots + 9 y^{2} z^{6} $
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Geometric Weierstrass model Geometric Weierstrass model

$ w^{2} $ $=$ $ -3 x^{3} z + 3 x z^{3} $
$0$$=$ $x^{2} + y^{2} + z^{2}$
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Rational points

This modular curve has no real points, and therefore no 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{2^2}{3^2}\cdot\frac{19970260992xw^{11}-709995331584xw^{10}u-1455989686272xw^{9}u^{2}+8435273416704xw^{8}u^{3}+13519493443584xw^{7}u^{4}-22655359328256xw^{6}u^{5}-33850562079744xw^{5}u^{6}+43768378791936xw^{4}u^{7}+22002960082176xw^{3}u^{8}-158052751597440xw^{2}u^{9}-67349953207872xwu^{10}+69735783810960xu^{11}-47250210816yw^{11}+2404433756160yw^{9}u^{2}+151711875072yw^{8}u^{3}-15911289022464yw^{7}u^{4}-218730792960yw^{6}u^{5}+33112182730752yw^{5}u^{6}-11610497885184yw^{4}u^{7}-32445679749120yw^{3}u^{8}+80243332053696yw^{2}u^{9}-43179927977088ywu^{10}-122287616293248z^{2}u^{10}+74530160640zw^{11}-709995331584zw^{10}u-3446040526848zw^{9}u^{2}+8042162651136zw^{8}u^{3}+20003841368064zw^{7}u^{4}-18569377640448zw^{6}u^{5}-29386156188672zw^{5}u^{6}+20565628130304zw^{4}u^{7}-6146404146432zw^{3}u^{8}-38923785660672zw^{2}u^{9}+149773019527872zwu^{10}+26640576zt^{11}+165671082zt^{10}u-6593326776zt^{9}u^{2}-52618057974zt^{8}u^{3}+781273280340zt^{7}u^{4}+1237040654040zt^{6}u^{5}-9202569906636zt^{5}u^{6}-11016384504408zt^{4}u^{7}+4464232870212zt^{3}u^{8}+35897902853526zt^{2}u^{9}+175434569715852ztu^{10}-144587125088442zu^{11}-519752318976w^{11}u-50236194816w^{10}u^{2}+5822274650112w^{9}u^{3}+1875598073856w^{8}u^{4}-8257086314496w^{7}u^{5}-12495551007744w^{6}u^{6}-20836686799872w^{5}u^{7}+58303721068416w^{4}u^{8}+45212282241408w^{3}u^{9}-328143132w^{2}t^{10}+17809773624w^{2}t^{9}u-55121588172w^{2}t^{8}u^{2}-266813072064w^{2}t^{7}u^{3}+6127015726080w^{2}t^{6}u^{4}-18476069950656w^{2}t^{5}u^{5}-56663347903632w^{2}t^{4}u^{6}-133316682624w^{2}t^{3}u^{7}-22577164144452w^{2}t^{2}u^{8}+264601939961160w^{2}tu^{9}+209160652598460w^{2}u^{10}-135831474wt^{11}+20353799916wt^{10}u-98897065686wt^{9}u^{2}+247011613776wt^{8}u^{3}+4810574931876wt^{7}u^{4}-24483555979560wt^{6}u^{5}-25716429230148wt^{5}u^{6}-5448063375648wt^{4}u^{7}-87210838832418wt^{3}u^{8}+247296555408204wt^{2}u^{9}+219717790365258wtu^{10}-43179927977088wu^{11}+14103662t^{12}+5324148072t^{11}u-36140977215t^{10}u^{2}+232254106524t^{9}u^{3}+1007948531124t^{8}u^{4}-8133276653964t^{7}u^{5}-1912498243182t^{6}u^{6}-10327045963500t^{5}u^{7}-42967616683302t^{4}u^{8}+82419501312828t^{3}u^{9}+76933859930757t^{2}u^{10}-43179927977088tu^{11}-11943936u^{12}}{u^{4}(2488320xw^{7}-56318976xw^{6}u-91031040xw^{5}u^{2}+242175744xw^{4}u^{3}+349168320xw^{3}u^{4}-328224960xw^{2}u^{5}-760984144xwu^{6}+494745488xu^{7}-5889024yw^{7}+130761216yw^{5}u^{2}+622080yw^{4}u^{3}-350482464yw^{3}u^{4}+74315232yw^{2}u^{5}-245393952ywu^{6}-612292416z^{2}u^{6}+9289728zw^{7}-56318976zw^{6}u-176214528zw^{5}u^{2}+222331392zw^{4}u^{3}+340039296zw^{3}u^{4}-185659776zw^{2}u^{5}+374722576zwu^{6}+53136zt^{7}-192618zt^{6}u-10530000zt^{5}u^{2}+8143578zt^{4}u^{3}+54099252zt^{3}u^{4}+56449062zt^{2}u^{5}+604213612ztu^{6}-830765318zu^{7}-41223168w^{7}u-4105728w^{6}u^{2}+141274368w^{5}u^{3}+77060160w^{4}u^{4}+169602336w^{3}u^{5}+571244w^{2}t^{6}-10236392w^{2}t^{5}u-29531468w^{2}t^{4}u^{2}+325004784w^{2}t^{3}u^{3}+290745748w^{2}t^{2}u^{4}+1304210168w^{2}tu^{5}+145377852w^{2}u^{6}+431762wt^{7}-14013908wt^{6}u-2471150wt^{5}u^{2}+272767560wt^{4}u^{3}+125533774wt^{3}u^{4}+1474205516wt^{2}u^{5}+175586478wtu^{6}-245393952wu^{7}+73070t^{8}-4955888t^{7}u+5796327t^{6}u^{2}+66913288t^{5}u^{3}+16270874t^{4}u^{4}+522150332t^{3}u^{5}+91534615t^{2}u^{6}-245393952tu^{7})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.cj.1 :

$\displaystyle X$ $=$ $\displaystyle t$
$\displaystyle Y$ $=$ $\displaystyle 2w$
$\displaystyle Z$ $=$ $\displaystyle u$

Equation of the image curve:

$0$ $=$ $ 3X^{8}+10X^{7}Y+13X^{6}Y^{2}+8X^{5}Y^{3}+2X^{4}Y^{4}-2X^{7}Z-4X^{6}YZ-2X^{5}Y^{2}Z+28X^{6}Z^{2}+78X^{5}YZ^{2}+87X^{4}Y^{2}Z^{2}+48X^{3}Y^{3}Z^{2}+12X^{2}Y^{4}Z^{2}+4X^{5}Z^{3}+67X^{4}Z^{4}+150X^{3}YZ^{4}+147X^{2}Y^{2}Z^{4}+72XY^{3}Z^{4}+18Y^{4}Z^{4}+30X^{3}Z^{5}+36X^{2}YZ^{5}+18XY^{2}Z^{5}+18X^{2}Z^{6}+18XYZ^{6}+9Y^{2}Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.0-8.j.1.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.96.0-8.j.1.4 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.96.1-48.b.1.2 $48$ $2$ $2$ $1$ $0$ $2$
48.96.1-48.b.1.8 $48$ $2$ $2$ $1$ $0$ $2$
48.96.2-48.d.1.1 $48$ $2$ $2$ $2$ $0$ $1$
48.96.2-48.d.1.6 $48$ $2$ $2$ $2$ $0$ $1$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.384.5-48.v.2.2 $48$ $2$ $2$ $5$ $1$ $1^{2}$
48.384.5-48.ce.1.1 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.dz.1.3 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.el.2.1 $48$ $2$ $2$ $5$ $1$ $1^{2}$
48.576.19-48.mu.2.1 $48$ $3$ $3$ $19$ $1$ $1^{8}\cdot2^{2}\cdot4$
48.768.21-48.jv.2.2 $48$ $4$ $4$ $21$ $1$ $1^{8}\cdot2^{3}\cdot4$
240.384.5-240.tm.1.3 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ty.1.1 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.xu.1.3 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.yg.2.1 $240$ $2$ $2$ $5$ $?$ not computed