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Properties

Label 48.192.3-48.pw.1.13
Level $48$
Index $192$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $4$

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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: 48.192.3.5425

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}1&7\\12&13\end{bmatrix}$, $\begin{bmatrix}17&43\\24&47\end{bmatrix}$, $\begin{bmatrix}19&12\\12&25\end{bmatrix}$, $\begin{bmatrix}19&36\\12&41\end{bmatrix}$, $\begin{bmatrix}31&21\\24&29\end{bmatrix}$, $\begin{bmatrix}47&8\\36&41\end{bmatrix}$
Contains $-I$: no $\quad$ (see 48.96.3.pw.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: $2^{15}\cdot3^{3}$
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 y w + x y t + 2 y^{2} w + y z t $
$=$ $x w^{2} + x w t + 2 y w^{2} + z w t$
$=$ $x w t + x t^{2} + 2 y w t + z t^{2}$
$=$ $x^{2} w + x^{2} t + 2 x y w + x z t$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 8 x^{7} + 28 x^{6} z + 38 x^{5} z^{2} + 30 x^{4} y^{2} z + 25 x^{4} z^{3} + 60 x^{3} y^{2} z^{2} + \cdots + 9 y^{4} z^{3} $
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Weierstrass model Weierstrass model

$ y^{2} + x^{4} y $ $=$ $ 2x^{8} - 10x^{4} + 4 $
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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
$(0:0:0:-1:1)$, $(1:-1/2:1:0:0)$, $(0:0:1:0:0)$, $(-1:1/2:1:0:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^3\,\frac{305783424y^{2}z^{12}-1665684000y^{2}z^{8}t^{4}+14268633048y^{2}z^{4}t^{8}-38263750y^{2}t^{12}+153031680yz^{13}-39331008yz^{11}wt-39331008yz^{11}t^{2}-27130464yz^{9}w^{2}t^{2}-54260928yz^{9}wt^{3}+969164352yz^{9}t^{4}+2670288768yz^{7}wt^{5}+2670288768yz^{7}t^{6}+1439310600yz^{5}w^{2}t^{6}+2878621200yz^{5}wt^{7}-8536938912yz^{5}t^{8}-2283108660yz^{3}wt^{9}-2283108660yz^{3}t^{10}+133856352yzw^{2}t^{10}+267712704yzwt^{11}+38263396yzt^{12}+93312z^{14}-25287552z^{12}wt-25287552z^{12}t^{2}-6010848z^{10}w^{2}t^{2}-12021696z^{10}wt^{3}-1744416z^{10}t^{4}-151479936z^{8}wt^{5}-151479936z^{8}t^{6}-424641600z^{6}w^{2}t^{6}-849283200z^{6}wt^{7}+1488792744z^{6}t^{8}+982337016z^{4}wt^{9}+982337016z^{4}t^{10}+277367562z^{2}w^{2}t^{10}+554735124z^{2}wt^{11}-28676032z^{2}t^{12}-1594323wt^{13}-1594323t^{14}}{tz(19440y^{2}z^{7}t^{3}-34308y^{2}z^{3}t^{7}-15552yz^{10}w-15552yz^{10}t-2592yz^{8}w^{2}t-5184yz^{8}wt^{2}-7776yz^{8}t^{3}-4752yz^{6}wt^{4}-4752yz^{6}t^{5}-9504yz^{4}w^{2}t^{5}-19008yz^{4}wt^{6}-18288yz^{4}t^{7}+1518yz^{2}wt^{8}+1518yz^{2}t^{9}+77yw^{2}t^{9}+154ywt^{10}-4yt^{11}+2592z^{9}w^{2}t+5184z^{9}wt^{2}-2592z^{9}t^{3}-1728z^{7}wt^{4}-1728z^{7}t^{5}-5220z^{5}w^{2}t^{5}-10440z^{5}wt^{6}+5724z^{5}t^{7}+1266z^{3}wt^{8}+1266z^{3}t^{9}+50zw^{2}t^{9}+100zwt^{10}-31zt^{11})}$

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

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

Equation of the image curve:

$0$ $=$ $ 8X^{7}+28X^{6}Z+30X^{4}Y^{2}Z+38X^{5}Z^{2}+60X^{3}Y^{2}Z^{2}+18XY^{4}Z^{2}+25X^{4}Z^{3}+24X^{2}Y^{2}Z^{3}+9Y^{4}Z^{3}+8X^{3}Z^{4}-6XY^{2}Z^{4}+X^{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.1 :

$\displaystyle X$ $=$ $\displaystyle -\frac{1}{2}w^{6}-3w^{5}t-\frac{13}{2}w^{4}t^{2}-6w^{3}t^{3}-2w^{2}t^{4}$
$\displaystyle Y$ $=$ $\displaystyle \frac{3}{2}z^{2}w^{21}t+\frac{63}{2}z^{2}w^{20}t^{2}+\frac{609}{2}z^{2}w^{19}t^{3}+\frac{3591}{2}z^{2}w^{18}t^{4}+\frac{14427}{2}z^{2}w^{17}t^{5}+\frac{41769}{2}z^{2}w^{16}t^{6}+\frac{89859}{2}z^{2}w^{15}t^{7}+\frac{145917}{2}z^{2}w^{14}t^{8}+89859z^{2}w^{13}t^{9}+83538z^{2}w^{12}t^{10}+57708z^{2}w^{11}t^{11}+28728z^{2}w^{10}t^{12}+9744z^{2}w^{9}t^{13}+2016z^{2}w^{8}t^{14}+192z^{2}w^{7}t^{15}+\frac{1}{8}w^{24}+3w^{23}t+\frac{1063}{32}w^{22}t^{2}+\frac{3597}{16}w^{21}t^{3}+\frac{33217}{32}w^{20}t^{4}+\frac{27639}{8}w^{19}t^{5}+\frac{272953}{32}w^{18}t^{6}+\frac{252957}{16}w^{17}t^{7}+\frac{702187}{32}w^{16}t^{8}+\frac{44661}{2}w^{15}t^{9}+\frac{31435}{2}w^{14}t^{10}+6336w^{13}t^{11}-127w^{12}t^{12}-1896w^{11}t^{13}-1168w^{10}t^{14}-336w^{9}t^{15}-40w^{8}t^{16}$
$\displaystyle Z$ $=$ $\displaystyle 12z^{3}wt^{2}+12z^{3}t^{3}+\frac{7}{2}zw^{4}t+14zw^{3}t^{2}+13zw^{2}t^{3}-2zwt^{4}$

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.1.45 $24$ $2$ $2$ $1$ $0$ $2$
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.1.10 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.kr.3.4 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.kt.1.21 $48$ $2$ $2$ $5$ $1$ $1^{2}$
48.384.5-48.kt.3.18 $48$ $2$ $2$ $5$ $1$ $1^{2}$
48.384.5-48.kz.1.6 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.kz.2.4 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.lb.1.19 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.5-48.lb.2.18 $48$ $2$ $2$ $5$ $0$ $1^{2}$
48.384.7-48.et.1.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.et.3.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.eu.2.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.eu.4.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ev.2.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ev.4.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ew.1.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ew.3.1 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ff.1.7 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ff.2.6 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fg.1.6 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fg.2.6 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fh.2.13 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fh.4.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fi.2.11 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fi.4.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fj.2.13 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fj.4.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fk.2.10 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fk.4.11 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fl.1.7 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fl.2.6 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fm.1.6 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fm.2.6 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fr.1.5 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fr.2.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fs.2.9 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fs.4.2 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ft.2.9 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.ft.4.3 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fu.1.5 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.7-48.fu.2.3 $48$ $2$ $2$ $7$ $0$ $2^{2}$
48.384.9-48.hq.2.42 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.iz.2.17 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.mj.1.19 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.mp.2.21 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.bag.1.19 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bag.2.18 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bai.1.6 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot2$
48.384.9-48.bai.2.4 $48$ $2$ $2$ $9$ $0$ $1^{4}\cdot2$
48.384.9-48.beg.2.22 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.bei.2.17 $48$ $2$ $2$ $9$ $1$ $1^{2}\cdot4$
48.384.9-48.bek.2.13 $48$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
48.384.9-48.bem.2.25 $48$ $2$ $2$ $9$ $1$ $1^{2}\cdot4$
48.384.9-48.bew.1.21 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
48.384.9-48.bew.3.18 $48$ $2$ $2$ $9$ $2$ $1^{4}\cdot2$
48.384.9-48.bey.1.10 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.384.9-48.bey.3.4 $48$ $2$ $2$ $9$ $1$ $1^{4}\cdot2$
48.576.13-48.co.1.5 $48$ $3$ $3$ $13$ $0$ $1^{4}\cdot2^{3}$
240.384.5-240.ckc.1.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckc.2.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckd.1.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckd.2.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cks.1.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cks.2.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckt.1.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.ckt.2.4 $240$ $2$ $2$ $5$ $?$ not computed
240.384.7-240.qv.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qv.3.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qw.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qw.3.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qx.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qx.3.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qy.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.qy.3.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rh.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rh.2.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ri.1.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ri.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rj.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rj.3.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rk.1.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rk.3.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rl.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rl.3.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rm.1.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rm.3.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rn.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rn.2.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ro.1.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ro.2.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rt.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rt.2.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ru.1.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.ru.3.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rv.1.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rv.3.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rw.1.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.rw.2.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.9-240.fpb.2.26 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpc.2.3 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpf.2.26 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpg.2.19 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpr.1.4 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fpr.2.4 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fps.1.7 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fps.2.7 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.frn.2.26 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fro.2.5 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.frr.2.26 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.frs.2.37 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fsd.1.4 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fsd.2.4 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fse.1.7 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.fse.2.7 $240$ $2$ $2$ $9$ $?$ not computed