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Properties

Label 48.192.3-48.qg.1.2
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 $1^{4}\cdot3^{4}\cdot4\cdot12\cdot16\cdot48$ Cusp orbits $1^{4}\cdot2^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 48J3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.192.3.5195

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}19&33\\0&29\end{bmatrix}$, $\begin{bmatrix}19&34\\36&17\end{bmatrix}$, $\begin{bmatrix}31&13\\12&43\end{bmatrix}$, $\begin{bmatrix}35&22\\24&23\end{bmatrix}$, $\begin{bmatrix}37&7\\0&47\end{bmatrix}$, $\begin{bmatrix}43&46\\0&11\end{bmatrix}$
Contains $-I$: no $\quad$ (see 48.96.3.qg.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 w + x y t + 2 y^{2} w - y z t $
$=$ $x w t + x t^{2} + 2 y w t - z t^{2}$
$=$ $x w^{2} + x w t + 2 y w^{2} - z w t$
$=$ $x z w + x z t + 2 y z w - z^{2} t$
$=$$\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 - 6 y^{2} z^{5} $
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Weierstrass model Weierstrass model

$ y^{2} + y $ $=$ $ 4x^{8} + 126x^{4} + 20 $
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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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^3\,\frac{305783424y^{2}z^{12}+899359200y^{2}z^{8}t^{4}+62445528y^{2}z^{4}t^{8}-52486y^{2}t^{12}+153031680yz^{13}+497959488yz^{11}wt+497959488yz^{11}t^{2}+217720224yz^{9}w^{2}t^{2}+435440448yz^{9}wt^{3}+993648384yz^{9}t^{4}-115883136yz^{7}wt^{5}-115883136yz^{7}t^{6}-10427112yz^{5}w^{2}t^{6}-20854224yz^{5}wt^{7}+17938368yz^{5}t^{8}-696900yz^{3}wt^{9}-696900yz^{3}t^{10}+215124yzw^{2}t^{10}+430248yzwt^{11}+52144yzt^{12}+93312z^{14}+25287552z^{12}wt+25287552z^{12}t^{2}+70427232z^{10}w^{2}t^{2}+140854464z^{10}wt^{3}+243106272z^{10}t^{4}+19678464z^{8}wt^{5}+19678464z^{8}t^{6}-19794816z^{6}w^{2}t^{6}-39589632z^{6}wt^{7}-24030792z^{6}t^{8}+2329800z^{4}wt^{9}+2329800z^{4}t^{10}-313194z^{2}w^{2}t^{10}-626388z^{2}wt^{11}-449932z^{2}t^{12}+2187wt^{13}+2187t^{14}}{t^{4}z(4252176y^{2}z^{7}+287172y^{2}z^{3}t^{4}+2125440yz^{8}+2664144yz^{6}wt+2664144yz^{6}t^{2}+174528yz^{4}w^{2}t^{2}+349056yz^{4}wt^{3}+571968yz^{4}t^{4}+5718yz^{2}wt^{5}+5718yz^{2}t^{6}-39yw^{2}t^{6}-78ywt^{7}+4yt^{8}+350784z^{7}wt+350784z^{7}t^{2}+277236z^{5}w^{2}t^{2}+554472z^{5}wt^{3}+448740z^{5}t^{4}+4242z^{3}wt^{5}+4242z^{3}t^{6}+90zw^{2}t^{6}+180zwt^{7}+133zt^{8})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 48.96.3.qg.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}-60X^{2}Y^{2}Z^{3}+9Y^{4}Z^{3}+8X^{3}Z^{4}-30XY^{2}Z^{4}+X^{2}Z^{5}-6Y^{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.qg.1 :

$\displaystyle X$ $=$ $\displaystyle 8z^{3}wt^{2}+8z^{3}t^{3}-\frac{7}{3}zw^{4}t-\frac{28}{3}zw^{3}t^{2}-\frac{50}{3}zw^{2}t^{3}-\frac{44}{3}zwt^{4}-\frac{16}{3}zt^{5}-\frac{1}{6}w^{5}t-\frac{5}{6}w^{4}t^{2}-\frac{14}{9}w^{3}t^{3}-\frac{4}{3}w^{2}t^{4}-\frac{4}{9}wt^{5}$
$\displaystyle Y$ $=$ $\displaystyle -\frac{416}{81}z^{3}w^{19}t^{2}-\frac{23920}{243}z^{3}w^{18}t^{3}-\frac{71696}{81}z^{3}w^{17}t^{4}-\frac{1205704}{243}z^{3}w^{16}t^{5}-\frac{524480}{27}z^{3}w^{15}t^{6}-\frac{41089888}{729}z^{3}w^{14}t^{7}-\frac{274421456}{2187}z^{3}w^{13}t^{8}-\frac{1435204600}{6561}z^{3}w^{12}t^{9}-\frac{660381664}{2187}z^{3}w^{11}t^{10}-\frac{724540544}{2187}z^{3}w^{10}t^{11}-\frac{1889893312}{6561}z^{3}w^{9}t^{12}-\frac{429640192}{2187}z^{3}w^{8}t^{13}-\frac{75126784}{729}z^{3}w^{7}t^{14}-\frac{263793152}{6561}z^{3}w^{6}t^{15}-\frac{8022016}{729}z^{3}w^{5}t^{16}-\frac{458752}{243}z^{3}w^{4}t^{17}-\frac{1003520}{6561}z^{3}w^{3}t^{18}-\frac{8}{9}z^{2}w^{21}t-\frac{176}{9}z^{2}w^{20}t^{2}-\frac{16346}{81}z^{2}w^{19}t^{3}-\frac{105158}{81}z^{2}w^{18}t^{4}-\frac{52520}{9}z^{2}w^{17}t^{5}-\frac{58388}{3}z^{2}w^{16}t^{6}-\frac{12127054}{243}z^{2}w^{15}t^{7}-\frac{24434186}{243}z^{2}w^{14}t^{8}-\frac{352670276}{2187}z^{2}w^{13}t^{9}-\frac{452973824}{2187}z^{2}w^{12}t^{10}-\frac{466168720}{2187}z^{2}w^{11}t^{11}-\frac{382356448}{2187}z^{2}w^{10}t^{12}-\frac{82321792}{729}z^{2}w^{9}t^{13}-\frac{41004160}{729}z^{2}w^{8}t^{14}-\frac{45639424}{2187}z^{2}w^{7}t^{15}-\frac{11886592}{2187}z^{2}w^{6}t^{16}-\frac{1940480}{2187}z^{2}w^{5}t^{17}-\frac{149504}{2187}z^{2}w^{4}t^{18}+\frac{148}{243}zw^{22}t+\frac{9842}{729}zw^{21}t^{2}+\frac{35096}{243}zw^{20}t^{3}+\frac{240965}{243}zw^{19}t^{4}+\frac{396569}{81}zw^{18}t^{5}+\frac{40334555}{2187}zw^{17}t^{6}+\frac{359983249}{6561}zw^{16}t^{7}+\frac{2591255930}{19683}zw^{15}t^{8}+\frac{564333458}{2187}zw^{14}t^{9}+\frac{8190021076}{19683}zw^{13}t^{10}+\frac{10894758800}{19683}zw^{12}t^{11}+\frac{11942824976}{19683}zw^{11}t^{12}+\frac{10735353056}{19683}zw^{10}t^{13}+\frac{2613181184}{6561}zw^{9}t^{14}+\frac{169680512}{729}zw^{8}t^{15}+\frac{2093275904}{19683}zw^{7}t^{16}+\frac{721017856}{19683}zw^{6}t^{17}+\frac{176204800}{19683}zw^{5}t^{18}+\frac{27252736}{19683}zw^{4}t^{19}+\frac{2007040}{19683}zw^{3}t^{20}+\frac{7}{81}w^{24}+\frac{175}{81}w^{23}t+\frac{75133}{2916}w^{22}t^{2}+\frac{142213}{729}w^{21}t^{3}+\frac{18412505}{17496}w^{20}t^{4}+\frac{18816697}{4374}w^{19}t^{5}+\frac{20164589}{1458}w^{18}t^{6}+\frac{156636401}{4374}w^{17}t^{7}+\frac{11942546021}{157464}w^{16}t^{8}+\frac{2611030690}{19683}w^{15}t^{9}+\frac{11373620182}{59049}w^{14}t^{10}+\frac{13729528544}{59049}w^{13}t^{11}+\frac{1527793664}{6561}w^{12}t^{12}+\frac{11359292560}{59049}w^{11}t^{13}+\frac{7663026032}{59049}w^{10}t^{14}+\frac{17100160}{243}w^{9}t^{15}+\frac{1768346240}{59049}w^{8}t^{16}+\frac{569065984}{59049}w^{7}t^{17}+\frac{43417600}{19683}w^{6}t^{18}+\frac{18904064}{59049}w^{5}t^{19}+\frac{1308160}{59049}w^{4}t^{20}$
$\displaystyle Z$ $=$ $\displaystyle -\frac{8}{3}z^{3}wt^{2}-\frac{8}{3}z^{3}t^{3}+\frac{7}{9}zw^{4}t+\frac{28}{9}zw^{3}t^{2}+\frac{50}{9}zw^{2}t^{3}+\frac{44}{9}zwt^{4}+\frac{16}{9}zt^{5}+\frac{1}{3}w^{6}+2w^{5}t+\frac{43}{9}w^{4}t^{2}+\frac{52}{9}w^{3}t^{3}+\frac{32}{9}w^{2}t^{4}+\frac{8}{9}wt^{5}$

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.1.18 $24$ $2$ $2$ $1$ $0$ $1^{2}$
48.48.0-48.h.1.3 $48$ $4$ $4$ $0$ $0$ full Jacobian
48.96.1-24.ir.1.17 $48$ $2$ $2$ $1$ $0$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
48.384.5-48.ks.1.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-48.ks.2.5 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-48.ks.3.10 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-48.ks.4.11 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-48.kt.1.17 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-48.kt.2.17 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-48.kt.3.18 $48$ $2$ $2$ $5$ $1$ $2$
48.384.5-48.kt.4.19 $48$ $2$ $2$ $5$ $1$ $2$
48.384.7-48.gn.1.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gn.2.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gn.3.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gn.4.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.go.1.2 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.go.2.3 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.go.3.3 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.go.4.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gx.1.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gx.2.6 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gy.1.11 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gy.2.9 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gz.1.3 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.gz.2.4 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ha.1.10 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.ha.2.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hb.1.11 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hb.2.3 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hc.1.2 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hc.2.6 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hd.1.13 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hd.2.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.he.1.3 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.he.2.10 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hh.1.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hh.2.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hh.3.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hh.4.1 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hi.1.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hi.2.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hi.3.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.7-48.hi.4.5 $48$ $2$ $2$ $7$ $1$ $2^{2}$
48.384.9-48.ho.1.7 $48$ $2$ $2$ $9$ $3$ $1^{6}$
48.384.9-48.jb.1.5 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.mh.1.12 $48$ $2$ $2$ $9$ $3$ $1^{6}$
48.384.9-48.mq.1.1 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.zp.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.zt.1.3 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.zw.1.1 $48$ $2$ $2$ $9$ $2$ $1^{6}$
48.384.9-48.bih.1.17 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bih.2.17 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bih.3.21 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bih.4.21 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bii.1.5 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bii.2.5 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bii.3.13 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.384.9-48.bii.4.13 $48$ $2$ $2$ $9$ $1$ $2\cdot4$
48.576.13-48.bg.1.11 $48$ $3$ $3$ $13$ $3$ $1^{10}$
240.384.5-240.cmo.1.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cmo.2.5 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cmo.3.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cmo.4.5 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cmp.1.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cmp.2.5 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cmp.3.7 $240$ $2$ $2$ $5$ $?$ not computed
240.384.5-240.cmp.4.5 $240$ $2$ $2$ $5$ $?$ not computed
240.384.7-240.bcb.1.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcb.2.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcb.3.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcb.4.3 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcc.1.5 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcc.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcc.3.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcc.4.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcl.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcl.2.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcm.1.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcm.2.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcn.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcn.2.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bco.1.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bco.2.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcp.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcp.2.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcq.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcq.2.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcr.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcr.2.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcs.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcs.2.2 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcv.1.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcv.2.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcv.3.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcv.4.1 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcw.1.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcw.2.9 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcw.3.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.7-240.bcw.4.17 $240$ $2$ $2$ $7$ $?$ not computed
240.384.9-240.gad.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gae.1.17 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gah.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gai.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gat.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gau.1.33 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gax.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gay.1.1 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.ghn.1.3 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.ghn.2.3 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.ghn.3.19 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.ghn.4.19 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gho.1.3 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gho.2.3 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gho.3.11 $240$ $2$ $2$ $9$ $?$ not computed
240.384.9-240.gho.4.11 $240$ $2$ $2$ $9$ $?$ not computed