↑

Properties

Label 48.192.3-48.pw.2.25
Level $48$
Index $192$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $48$ $\SL_2$-level: $48$ Newform level: $192$
Index: $192$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $4$ are rational) Cusp widths $1^{4}\cdot3^{4}\cdot4\cdot12\cdot16\cdot48$ Cusp orbits $1^{4}\cdot2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 48I3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.192.3.5489

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&24\\12&19\end{bmatrix}$, $\begin{bmatrix}17&8\\0&25\end{bmatrix}$, $\begin{bmatrix}19&27\\12&11\end{bmatrix}$, $\begin{bmatrix}19&30\\12&29\end{bmatrix}$, $\begin{bmatrix}23&4\\12&37\end{bmatrix}$, $\begin{bmatrix}25&29\\36&13\end{bmatrix}$
Contains $-I$: no $\quad$ (see 48.96.3.pw.2 for the level structure with $-I$)
Cyclic 48-isogeny field degree: $2$
Cyclic 48-torsion field degree: $16$
Full 48-torsion field degree: $6144$

Jacobian

Conductor: $2^{15}\cdot3^{3}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 24.2.a.a, 192.2.c.a

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

$ 0 $ $=$ $ x^{2} w + x y w + x y t - y z w - y z t $
$=$ $x^{2} w - 2 x y w + x y t + 2 y^{2} w - 2 y^{2} t + y z w$
$=$ $x^{2} w - x y w - 2 x z w - x z t - 2 y^{2} w + 2 y^{2} t + y z t$
$=$ $3 x z w - 2 y z w + 2 y z t - 2 z^{2} w - z^{2} t$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 50 x^{7} - 80 x^{6} z - 30 x^{5} y^{2} + 12 x^{5} z^{2} + 129 x^{4} y^{2} z + 16 x^{4} z^{3} + \cdots - 3 y^{2} z^{5} $
 Copy content Toggle raw display

Weierstrass model Weierstrass model

$ y^{2} + x^{4} y $ $=$ $ -10x^{4} + 36 $
 Copy content Toggle raw display

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:0:0:1)$, $(-1:-1/2:1:0:0)$, $(0:0:1:0:0)$, $(1:1/2:1:0:0)$

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{2^3}{3}\cdot\frac{704692221113580650496xzt^{12}+3339970412544000000000000y^{2}z^{12}+148944863232000000000000y^{2}z^{8}t^{4}+6071086080000000000000y^{2}z^{6}t^{6}-30216715848000000000000y^{2}z^{4}t^{8}-35322541591500000000000y^{2}z^{2}t^{10}-12480619028588793507672y^{2}t^{12}+1671514030080000000000000yz^{13}+123382849536000000000000yz^{9}t^{4}+41337506304000000000000yz^{7}t^{6}+15590829348000000000000yz^{5}t^{8}+5650993131750000000000yz^{3}t^{10}+9661850132588793507672yzt^{12}+1019215872000000000000z^{14}+25894453248000000000000z^{10}t^{4}+19420839936000000000000z^{8}t^{6}+15489449856000000000000z^{6}t^{8}+11661683400000000000000z^{4}t^{10}+170185248243600000000000z^{2}w^{12}+754791219418080000000000z^{2}w^{11}t+1158711885052584000000000z^{2}w^{10}t^{2}+312125799687883200000000z^{2}w^{9}t^{3}-1220019528646226640000000z^{2}w^{8}t^{4}-1719086945184354672000000z^{2}w^{7}t^{5}-1277098023593816565600000z^{2}w^{6}t^{6}-536978108883899186880000z^{2}w^{5}t^{7}-42680636509395162624000z^{2}w^{4}t^{8}-88611631794039717475200z^{2}w^{3}t^{9}+20935328749489193495040z^{2}w^{2}t^{10}+20683092854195911300992z^{2}wt^{11}-376210388073279027414z^{2}t^{12}-12216462411500000000000w^{14}+51632780236750000000000w^{13}t+281239665256875000000000w^{12}t^{2}+268317161164312500000000w^{11}t^{3}-272437005789218750000000w^{10}t^{4}-660261329162578125000000w^{9}t^{5}-297222120043257812500000w^{8}t^{6}+186808540701535156250000w^{7}t^{7}+251577143903419921875000w^{6}t^{8}+162972945994329101562500w^{5}t^{9}+29803282286179199218750w^{4}t^{10}+10819942057246337890625w^{3}t^{11}+503635968481069891584w^{2}t^{12}+23887872000000000000wt^{13}}{2654207989128364032xzt^{12}-1327104000000000000y^{2}z^{8}t^{4}-248832000000000000y^{2}z^{6}t^{6}-75384000000000000y^{2}z^{4}t^{8}-43420500000000000y^{2}z^{2}t^{10}-3550740008636139624y^{2}t^{12}+663552000000000000yz^{9}t^{4}+124416000000000000yz^{7}t^{6}+23868000000000000yz^{5}t^{8}+5870250000000000yz^{3}t^{10}-7066091991363860376yzt^{12}+10368000000000000z^{6}t^{8}+9720000000000000z^{4}t^{10}-1269334800000000000z^{2}w^{12}-20674556640000000000z^{2}w^{11}t-16747293672000000000z^{2}w^{10}t^{2}+98970347534400000000z^{2}w^{9}t^{3}+176034869943120000000z^{2}w^{8}t^{4}+24341418011376000000z^{2}w^{7}t^{5}-172257916614775200000z^{2}w^{6}t^{6}-191810318714544960000z^{2}w^{5}t^{7}-98751114282091008000z^{2}w^{4}t^{8}-26470320074831798400z^{2}w^{3}t^{9}-374333516283640320z^{2}w^{2}t^{10}+4759035259506728064z^{2}wt^{11}+1744021321212601062z^{2}t^{12}-609808500000000000w^{14}+1742249250000000000w^{13}t+7503202125000000000w^{12}t^{2}-1721639812500000000w^{11}t^{3}-16062946531250000000w^{10}t^{4}-8260774046875000000w^{9}t^{5}+6081213445312500000w^{8}t^{6}+7921711121093750000w^{7}t^{7}+3427921751953125000w^{6}t^{8}+781692022460937500w^{5}t^{9}+75215895019531250w^{4}t^{10}+8173779052734375w^{3}t^{11}-442367998188060672w^{2}t^{12}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.pw.2 :

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

Equation of the image curve:

$0$ $=$ $ 50X^{7}-30X^{5}Y^{2}+72X^{3}Y^{4}-80X^{6}Z+129X^{4}Y^{2}Z+72X^{2}Y^{4}Z+12X^{5}Z^{2}+12X^{3}Y^{2}Z^{2}+18XY^{4}Z^{2}+16X^{4}Z^{3}-78X^{2}Y^{2}Z^{3}+2X^{3}Z^{4}-30XY^{2}Z^{4}-3Y^{2}Z^{5} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 48.96.3.pw.2 :

$\displaystyle X$ $=$ $\displaystyle -\frac{1}{4}w^{6}+\frac{2}{5}w^{5}t-\frac{3}{50}w^{4}t^{2}-\frac{2}{25}w^{3}t^{3}-\frac{1}{100}w^{2}t^{4}$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{80}z^{2}w^{22}-\frac{51}{800}z^{2}w^{21}t+\frac{231}{2000}z^{2}w^{20}t^{2}-\frac{259}{4000}z^{2}w^{19}t^{3}-\frac{483}{10000}z^{2}w^{18}t^{4}+\frac{31017}{500000}z^{2}w^{17}t^{5}+\frac{5887}{1250000}z^{2}w^{16}t^{6}-\frac{265203}{12500000}z^{2}w^{15}t^{7}-\frac{7569}{6250000}z^{2}w^{14}t^{8}+\frac{10339}{2500000}z^{2}w^{13}t^{9}+\frac{5313}{6250000}z^{2}w^{12}t^{10}-\frac{777}{2500000}z^{2}w^{11}t^{11}-\frac{1001}{6250000}z^{2}w^{10}t^{12}-\frac{357}{12500000}z^{2}w^{9}t^{13}-\frac{3}{1250000}z^{2}w^{8}t^{14}-\frac{1}{12500000}z^{2}w^{7}t^{15}-\frac{1}{384}w^{24}+\frac{37}{1920}w^{23}t-\frac{749}{12800}w^{22}t^{2}+\frac{269}{3000}w^{21}t^{3}-\frac{299}{4800}w^{20}t^{4}-\frac{413}{100000}w^{19}t^{5}+\frac{192367}{6000000}w^{18}t^{6}-\frac{2266}{234375}w^{17}t^{7}-\frac{98493}{12500000}w^{16}t^{8}+\frac{78309}{25000000}w^{15}t^{9}+\frac{484451}{300000000}w^{14}t^{10}-\frac{1261}{3125000}w^{13}t^{11}-\frac{20657}{75000000}w^{12}t^{12}-\frac{287}{37500000}w^{11}t^{13}+\frac{1117}{50000000}w^{10}t^{14}+\frac{31}{4687500}w^{9}t^{15}+\frac{133}{150000000}w^{8}t^{16}+\frac{3}{50000000}w^{7}t^{17}+\frac{1}{600000000}w^{6}t^{18}$
$\displaystyle Z$ $=$ $\displaystyle -\frac{24}{25}z^{3}w^{3}-\frac{24}{25}z^{3}w^{2}t-\frac{6}{25}z^{3}wt^{2}+\frac{3}{10}zw^{5}-\frac{49}{100}zw^{4}t-\frac{4}{25}zw^{3}t^{2}+\frac{6}{25}zw^{2}t^{3}+\frac{1}{10}zwt^{4}+\frac{1}{100}zt^{5}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
24.96.1-24.ir.1.37 $24$ $2$ $2$ $1$ $0$ $2$
48.96.1-24.ir.1.17 $48$ $2$ $2$ $1$ $0$ $2$

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.kr.2.11 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.kr.4.7 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.kt.2.21 $48$ $2$ $2$ $5$ $1$ $1^{2}$
48.384.5-48.kt.4.19 $48$ $2$ $2$ $5$ $1$ $1^{2}$
48.384.5-48.kz.3.10 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.kz.4.4 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.lb.3.21 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.lb.4.18 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.7-48.et.2.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.et.4.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.eu.1.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.eu.3.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ev.1.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ev.3.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ew.2.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ew.4.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ff.3.13 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ff.4.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fg.3.11 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fg.4.11 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fh.1.13 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fh.3.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fi.1.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fi.3.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fj.1.13 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fj.3.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fk.1.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fk.3.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fl.3.13 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fl.4.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fm.3.11 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fm.4.11 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fr.3.9 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fr.4.3 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fs.1.9 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fs.3.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ft.1.9 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ft.3.3 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fu.3.9 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fu.4.5 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.9-48.hq.4.36 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.iz.1.17 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.mj.2.23 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.mp.1.21 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.bag.3.21 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bag.4.18 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bai.3.10 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot2$
48.384.9-48.bai.4.4 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot2$
48.384.9-48.beg.1.22 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.bei.1.17 $48$ $2$ $2$ $9$ $1$ $1^{2}\cdot4$
48.384.9-48.bek.1.13 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.bem.1.25 $48$ $2$ $2$ $9$ $1$ $1^{2}\cdot4$
48.384.9-48.bew.2.21 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
48.384.9-48.bew.4.19 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
48.384.9-48.bey.2.11 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bey.4.7 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.576.13-48.co.1.5 $48$ $3$ $3$ $13$ $0$ $1^{4}\cdot2^{3}$
240.384.5-240.ckc.3.13 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckc.4.13 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckd.3.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckd.4.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cks.3.13 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cks.4.11 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckt.3.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckt.4.6 $240$ $2$ $2$ $5$ $?$ not computed
240.384.7-240.qv.2.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qv.4.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qw.2.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qw.4.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qx.2.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qx.4.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qy.2.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qy.4.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rh.3.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rh.4.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ri.3.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ri.4.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rj.2.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rj.4.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rk.2.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rk.4.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rl.2.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rl.4.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rm.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rm.4.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rn.3.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rn.4.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ro.3.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ro.4.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rt.3.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rt.4.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ru.2.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ru.4.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rv.2.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rv.4.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rw.3.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rw.4.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.9-240.fpb.1.26 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpc.1.3 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpf.1.26 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpg.1.19 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpr.3.7 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpr.4.6 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fps.3.13 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fps.4.11 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.frn.1.26 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fro.1.5 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.frr.1.26 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.frs.1.37 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fsd.3.4 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fsd.4.4 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fse.3.7 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fse.4.7 $240$ $2$ $2$ $9$ $?$ not computed