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Properties

Label 48.192.3-24.fb.1.16
Level $48$
Index $192$
Genus $3$
Analytic rank $1$
Cusps $12$
$\Q$-cusps $4$

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Invariants

Level: $48$ $\SL_2$-level: $48$ 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: $12$ (of which $4$ are rational) Cusp widths $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ Cusp orbits $1^{4}\cdot2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 24V3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.192.3.6550

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}7&32\\24&11\end{bmatrix}$, $\begin{bmatrix}17&40\\0&11\end{bmatrix}$, $\begin{bmatrix}35&13\\24&23\end{bmatrix}$, $\begin{bmatrix}37&0\\0&13\end{bmatrix}$, $\begin{bmatrix}37&30\\0&7\end{bmatrix}$, $\begin{bmatrix}43&26\\24&25\end{bmatrix}$
Contains $-I$: no $\quad$ (see 24.96.3.fb.1 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: $2^{15}\cdot3^{5}$
Simple: no
Squarefree: yes
Decomposition: $1^{3}$
Newforms: 24.2.a.a, 576.2.a.b, 576.2.a.d

Models

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

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

$ 0 $ $=$ $ 4 x^{6} - 30 x^{4} y^{2} - 28 x^{4} z^{2} + 36 x^{2} y^{4} + 156 x^{2} y^{2} z^{2} + 57 x^{2} z^{4} + \cdots - 36 z^{6} $
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Weierstrass model Weierstrass model

$ y^{2} + x^{4} y $ $=$ $ 126x^{4} + 324 $
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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
$(0:0:0:-2:1)$, $(-1:-1:1:0:0)$, $(0:-2:1:0:0)$, $(-1:1:0:0:0)$

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{1}{3^2}\cdot\frac{268462703872896xzt^{12}+204073344y^{14}+22584116736y^{12}t^{2}+835181497728y^{10}t^{4}+10412517371904y^{8}t^{6}+3992173454592y^{6}t^{8}+24230438967744y^{4}t^{10}+73152134179488y^{2}t^{12}+3095384481792yz^{13}+74549897247744yz^{11}t^{2}+399132813745152yz^{9}t^{4}-389317097530368yz^{7}t^{6}-6290044173039744yz^{5}t^{8}-1080686364620544yz^{3}t^{10}-102765202833984yzt^{12}+3111710349312z^{14}+73819042578432z^{12}t^{2}+386207715128832z^{10}t^{4}-479880645198336z^{8}t^{6}-6312276956098944z^{6}t^{8}-226725198809472z^{4}t^{10}-2054238894z^{2}w^{12}+179531095356z^{2}w^{11}t+490674459294z^{2}w^{10}t^{2}+5799767689764z^{2}w^{9}t^{3}+13958781990660z^{2}w^{8}t^{4}+38520353290608z^{2}w^{7}t^{5}+96351707992878z^{2}w^{6}t^{6}-31409586096444z^{2}w^{5}t^{7}-121778289160848z^{2}w^{4}t^{8}-658926139607280z^{2}w^{3}t^{9}-1685013300371616z^{2}w^{2}t^{10}+1879506376192896z^{2}wt^{11}+4948056543091680z^{2}t^{12}+402780276w^{14}-29185207848w^{13}t-78797816712w^{12}t^{2}-895155928560w^{11}t^{3}-2139056507619w^{10}t^{4}-4247604922842w^{9}t^{5}-10724738134383w^{8}t^{6}+23036059841598w^{7}t^{7}+64180746897261w^{6}t^{8}+119163207710514w^{5}t^{9}+277031226659440w^{4}t^{10}-390523437710104w^{3}t^{11}-1033440205581408w^{2}t^{12}+282939716723856wt^{13}+802168879115232t^{14}}{t^{2}(17870275800xzt^{10}+3779136y^{12}-44089920y^{10}t^{2}+194730480y^{8}t^{4}-399433680y^{6}t^{6}+368637804y^{4}t^{8}-116361036y^{2}t^{10}+15475561920yz^{11}-125076174624yz^{9}t^{2}+324895366224yz^{7}t^{4}-334523817432yz^{5}t^{6}+112780539864yz^{3}t^{8}+68246921028yzt^{10}+15471782784z^{12}-127603786752z^{10}t^{2}+342903054240z^{8}t^{4}-373102269984z^{6}t^{6}+143895886992z^{4}t^{8}+1353979962z^{2}w^{9}t+3373681266z^{2}w^{8}t^{2}-20275421688z^{2}w^{7}t^{3}-50558540958z^{2}w^{6}t^{4}+128339259084z^{2}w^{5}t^{5}+319248962280z^{2}w^{4}t^{6}-396318427380z^{2}w^{3}t^{7}-982469628900z^{2}w^{2}t^{8}+510696543036z^{2}wt^{9}+1331689508424z^{2}t^{10}-225332604w^{11}t-561293388w^{10}t^{2}+3811519692w^{9}t^{3}+9495741024w^{8}t^{4}-26655079203w^{7}t^{5}-66224748627w^{6}t^{6}+92361679890w^{5}t^{7}+228453260199w^{4}t^{8}-148303512449w^{3}t^{9}-366990979897w^{2}t^{10}+86839833468wt^{11}+215942490852t^{12})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.fb.1 :

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

Equation of the image curve:

$0$ $=$ $ 4X^{6}-30X^{4}Y^{2}+36X^{2}Y^{4}-28X^{4}Z^{2}+156X^{2}Y^{2}Z^{2}-144Y^{4}Z^{2}+57X^{2}Z^{4}-240Y^{2}Z^{4}-36Z^{6} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 24.96.3.fb.1 :

$\displaystyle X$ $=$ $\displaystyle -\frac{6}{5}z^{3}w^{2}+\frac{24}{5}z^{3}t^{2}+\frac{7}{5}zw^{4}-\frac{37}{5}zw^{2}t^{2}+\frac{52}{5}zt^{4}+\frac{6}{5}w^{5}+\frac{12}{5}w^{4}t-5w^{3}t^{2}-10w^{2}t^{3}+\frac{24}{5}wt^{4}+\frac{48}{5}t^{5}$
$\displaystyle Y$ $=$ $\displaystyle \frac{39312}{625}z^{3}w^{17}+\frac{68256}{625}z^{3}w^{16}t-\frac{143208}{125}z^{3}w^{15}t^{2}-\frac{1247184}{625}z^{3}w^{14}t^{3}+\frac{5596812}{625}z^{3}w^{13}t^{4}+\frac{9782712}{625}z^{3}w^{12}t^{5}-\frac{4908046}{125}z^{3}w^{11}t^{6}-\frac{43058908}{625}z^{3}w^{10}t^{7}+\frac{66065792}{625}z^{3}w^{9}t^{8}+\frac{116400768}{625}z^{3}w^{8}t^{9}-\frac{22380384}{125}z^{3}w^{7}t^{10}-\frac{198026432}{625}z^{3}w^{6}t^{11}+\frac{116543232}{625}z^{3}w^{5}t^{12}+\frac{207189504}{625}z^{3}w^{4}t^{13}-\frac{13658112}{125}z^{3}w^{3}t^{14}-\frac{121982976}{625}z^{3}w^{2}t^{15}+\frac{17252352}{625}z^{3}wt^{16}+\frac{6193152}{125}z^{3}t^{17}+\frac{21024}{625}z^{2}w^{18}+\frac{8064}{125}z^{2}w^{17}t-\frac{75456}{125}z^{2}w^{16}t^{2}-\frac{145152}{125}z^{2}w^{15}t^{3}+\frac{2898096}{625}z^{2}w^{14}t^{4}+\frac{1119552}{125}z^{2}w^{13}t^{5}-\frac{37350544}{1875}z^{2}w^{12}t^{6}-\frac{14510272}{375}z^{2}w^{11}t^{7}+\frac{32709698}{625}z^{2}w^{10}t^{8}+\frac{7686728}{75}z^{2}w^{9}t^{9}-\frac{161263088}{1875}z^{2}w^{8}t^{10}-\frac{21320096}{125}z^{2}w^{7}t^{11}+\frac{10752224}{125}z^{2}w^{6}t^{12}+\frac{21788928}{125}z^{2}w^{5}t^{13}-\frac{29516544}{625}z^{2}w^{4}t^{14}-\frac{12515328}{125}z^{2}w^{3}t^{15}+\frac{6285312}{625}z^{2}w^{2}t^{16}+\frac{3096576}{125}z^{2}wt^{17}+\frac{442368}{625}z^{2}t^{18}-\frac{21672}{625}zw^{19}-\frac{45072}{625}zw^{18}t+\frac{78876}{125}zw^{17}t^{2}+\frac{843192}{625}zw^{16}t^{3}-\frac{3188634}{625}zw^{15}t^{4}-\frac{7008996}{625}zw^{14}t^{5}+\frac{45146731}{1875}zw^{13}t^{6}+\frac{101950142}{1875}zw^{12}t^{7}-\frac{137171363}{1875}zw^{11}t^{8}-\frac{63541814}{375}zw^{10}t^{9}+\frac{278352032}{1875}zw^{9}t^{10}+\frac{659655808}{1875}zw^{8}t^{11}-\frac{125735568}{625}zw^{7}t^{12}-\frac{911902432}{1875}zw^{6}t^{13}+\frac{109662336}{625}zw^{5}t^{14}+\frac{269530368}{625}zw^{4}t^{15}-\frac{11160576}{125}zw^{3}t^{16}-\frac{138958848}{625}zw^{2}t^{17}+\frac{12607488}{625}zwt^{18}+\frac{6340608}{125}zt^{19}-\frac{10074}{625}w^{20}-\frac{24576}{625}w^{19}t+\frac{174204}{625}w^{18}t^{2}+\frac{441792}{625}w^{17}t^{3}-\frac{1325599}{625}w^{16}t^{4}-\frac{3510656}{625}w^{15}t^{5}+\frac{52414343}{5625}w^{14}t^{6}+\frac{145686688}{5625}w^{13}t^{7}-\frac{1171186597}{45000}w^{12}t^{8}-\frac{429585184}{5625}w^{11}t^{9}+\frac{17940314}{375}w^{10}t^{10}+\frac{840041468}{5625}w^{9}t^{11}-\frac{324165562}{5625}w^{8}t^{12}-\frac{363083728}{1875}w^{7}t^{13}+\frac{81341744}{1875}w^{6}t^{14}+\frac{100328832}{625}w^{5}t^{15}-\frac{2227584}{125}w^{4}t^{16}-\frac{48239616}{625}w^{3}t^{17}+\frac{1453056}{625}w^{2}t^{18}+\frac{10248192}{625}wt^{19}+\frac{294912}{625}t^{20}$
$\displaystyle Z$ $=$ $\displaystyle \frac{6}{5}z^{3}w^{2}-\frac{24}{5}z^{3}t^{2}-\frac{7}{5}zw^{4}+\frac{37}{5}zw^{2}t^{2}-\frac{52}{5}zt^{4}-\frac{1}{5}w^{5}+\frac{2}{5}w^{4}t+\frac{5}{6}w^{3}t^{2}-\frac{5}{3}w^{2}t^{3}-\frac{4}{5}wt^{4}+\frac{8}{5}t^{5}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
48.48.0-24.bj.1.6 $48$ $4$ $4$ $0$ $0$ full Jacobian
48.96.1-24.ir.1.17 $48$ $2$ $2$ $1$ $0$ $1^{2}$
48.96.1-24.ir.1.23 $48$ $2$ $2$ $1$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.384.5-24.ey.1.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-24.ey.2.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-24.ey.3.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-24.ey.4.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-24.fa.1.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-24.fa.2.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-24.fa.3.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-24.fa.4.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.7-48.ee.1.17 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ee.1.25 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ee.2.17 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ee.2.25 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ef.1.19 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ef.1.23 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ef.2.19 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ef.2.23 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ef.3.21 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ef.3.22 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ef.4.21 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ef.4.22 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.eg.1.17 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.eg.1.21 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.eg.2.17 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.eg.2.21 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.ek.1.14 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.ek.1.16 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.ek.2.14 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.ek.2.16 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.el.1.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.el.1.2 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.el.2.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.el.2.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.em.1.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.em.1.3 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.em.2.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.em.2.3 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.en.1.14 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.en.1.16 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.en.2.14 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-24.en.2.16 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.9-48.zq.1.1 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.zq.1.5 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.zs.1.1 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.zs.1.9 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.beh.1.18 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.beh.1.26 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.beh.2.18 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.beh.2.26 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bei.1.17 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bei.1.21 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bei.2.17 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bei.2.21 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bgk.1.1 $48$ $2$ $2$ $9$ $3$ $1^{6}$
48.384.9-48.bgk.1.3 $48$ $2$ $2$ $9$ $3$ $1^{6}$
48.384.9-48.bgm.1.1 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.bgm.1.5 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.576.13-24.mm.1.4 $48$ $3$ $3$ $13$ $3$ $1^{10}$
240.384.5-120.vo.1.11 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.vo.2.11 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.vo.3.11 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.vo.4.11 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.vq.1.15 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.vq.2.15 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.vq.3.15 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-120.vq.4.15 $240$ $2$ $2$ $5$ $?$ not computed
240.384.7-120.ku.1.20 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.ku.1.28 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.ku.2.18 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.ku.2.26 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kv.1.6 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kv.1.14 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kv.2.25 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kv.2.27 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kw.1.10 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kw.1.12 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kw.2.21 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kw.2.29 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kx.1.20 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kx.1.28 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kx.2.18 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-120.kx.2.26 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qg.1.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qg.1.41 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qg.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qg.2.41 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qh.1.34 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qh.1.42 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qh.2.34 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qh.2.42 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qh.3.34 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qh.3.36 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qh.4.34 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qh.4.36 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qi.1.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qi.1.25 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qi.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qi.2.25 $240$ $2$ $2$ $7$ $?$ not computed
240.384.9-240.fkl.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fkl.1.9 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fkm.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fkm.1.17 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.flr.1.18 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.flr.1.50 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.flr.2.18 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.flr.2.50 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fls.1.5 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fls.1.21 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fls.2.5 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fls.2.21 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fnv.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fnv.1.9 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fnw.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fnw.1.17 $240$ $2$ $2$ $9$ $?$ not computed