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Properties

Label 112.384.11-56.fa.4.31
Level $112$
Index $384$
Genus $11$
Cusps $12$
$\Q$-cusps $6$

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Invariants

Level: $112$ $\SL_2$-level: $112$ Newform level: $56$
Index: $384$ $\PSL_2$-index:$192$
Genus: $11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $6$ are rational) Cusp widths $1^{2}\cdot2\cdot4\cdot7^{2}\cdot8^{2}\cdot14\cdot28\cdot56^{2}$ Cusp orbits $1^{6}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $4 \le \gamma \le 11$
$\overline{\Q}$-gonality: $4 \le \gamma \le 11$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 56O11

Level structure

$\GL_2(\Z/112\Z)$-generators: $\begin{bmatrix}3&80\\104&35\end{bmatrix}$, $\begin{bmatrix}19&90\\62&47\end{bmatrix}$, $\begin{bmatrix}28&79\\27&80\end{bmatrix}$, $\begin{bmatrix}37&86\\102&77\end{bmatrix}$, $\begin{bmatrix}71&44\\16&43\end{bmatrix}$, $\begin{bmatrix}104&73\\43&22\end{bmatrix}$
Contains $-I$: no $\quad$ (see 56.192.11.fa.4 for the level structure with $-I$)
Cyclic 112-isogeny field degree: $2$
Cyclic 112-torsion field degree: $48$
Full 112-torsion field degree: $129024$

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

$ 0 $ $=$ $ x u + x a - y u + z s + t v $
$=$ $x v + x r - w r - w a - w b$
$=$ $x u - x s + y u - w s$
$=$ $x r + x b + y u - y v - w v$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 73\!\cdots\!88 x^{18} y + \cdots - 145432 y^{6} z^{13} $
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Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(1:1:1:-3:-4:4:1:-7:-4:8:1)$, $(-1:-1:-1:3:4:4:1:-7:-4:8:1)$, $(0:0:0:0:0:0:-1:1:0:-2:1)$, $(0:0:0:0:0:0:-1/3:-1/3:0:0:1)$, $(0:0:0:0:0:4/7:-1/7:-5/7:0:-4/7:1)$, $(0:0:0:0:0:-2:-1:-3:-2:2:1)$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve $X_0(56)$ :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle -z$
$\displaystyle W$ $=$ $\displaystyle w$
$\displaystyle T$ $=$ $\displaystyle -t$

Equation of the image curve:

$0$ $=$ $ YZ+XW+XT $
$=$ $ X^{2}+Y^{2}-XZ-YW-2W^{2}-WT $
$=$ $ 2X^{2}-Y^{2}+XZ+Z^{2}-YW-YT $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.192.11.fa.4 :

$\displaystyle X$ $=$ $\displaystyle t-s$
$\displaystyle Y$ $=$ $\displaystyle u+s$
$\displaystyle Z$ $=$ $\displaystyle s+4b$

Equation of the image curve:

$0$ $=$ $ 73532736342130688X^{18}Y+804251670987735040X^{17}Y^{2}+4344773707528740864X^{16}Y^{3}+15401874376394354688X^{15}Y^{4}+40007722024971371776X^{14}Y^{5}+80569954210725668864X^{13}Y^{6}+129943469004737346560X^{12}Y^{7}+171110814149128906752X^{11}Y^{8}+186007263068915645952X^{10}Y^{9}+167733109698989764608X^{9}Y^{10}+125416214000590005248X^{8}Y^{11}+77300101387018739712X^{7}Y^{12}+38799237787574224128X^{6}Y^{13}+15541945961713690624X^{5}Y^{14}+4816300941722224640X^{4}Y^{15}+1102156451151529984X^{3}Y^{16}+173369973168758784X^{2}Y^{17}+16542843015544832XY^{18}+713887748374528Y^{19}-8650910157897728X^{18}Z-92047301072584704X^{17}YZ-510200549784289280X^{16}Y^{2}Z-1953576687145603072X^{15}Y^{3}Z-5693155576853552640X^{14}Y^{4}Z-13179987656634494976X^{13}Y^{5}Z-24761092288249189376X^{12}Y^{6}Z-38155259361404369408X^{11}Y^{7}Z-48466259691766312704X^{10}Y^{8}Z-50811834543753278464X^{9}Y^{9}Z-43886311411923672832X^{8}Y^{10}Z-31073108381112168448X^{7}Y^{11}Z-17872460935516416768X^{6}Y^{12}Z-8225410333069468672X^{5}Y^{13}Z-2952270044444921088X^{4}Y^{14}Z-791953266919933440X^{3}Y^{15}Z-148041839573010432X^{2}Y^{16}Z-17019503526402048XY^{17}Z-893552123703296Y^{18}Z-242548882931712X^{17}Z^{2}-362635946688512X^{16}YZ^{2}+12269848514101248X^{15}Y^{2}Z^{2}+95114415878673408X^{14}Y^{3}Z^{2}+381324791263947776X^{13}Y^{4}Z^{2}+1047009136212493888X^{12}Y^{5}Z^{2}+2173873258364666880X^{11}Y^{6}Z^{2}+3577649089516650400X^{10}Y^{7}Z^{2}+4766535377722465536X^{9}Y^{8}Z^{2}+5170709402694946432X^{8}Y^{9}Z^{2}+4551481373625190144X^{7}Y^{10}Z^{2}+3219326004459235104X^{6}Y^{11}Z^{2}+1801297256379552512X^{5}Y^{12}Z^{2}+778590419204575488X^{4}Y^{13}Z^{2}+249575162057720576X^{3}Y^{14}Z^{2}+55430897621473792X^{2}Y^{15}Z^{2}+7544050750387712XY^{16}Z^{2}+474772118499328Y^{17}Z^{2}+336729523945472X^{16}Z^{3}+3270281083453440X^{15}YZ^{3}+16216581907068928X^{14}Y^{2}Z^{3}+53058147734875136X^{13}Y^{3}Z^{3}+124963000396662400X^{12}Y^{4}Z^{3}+220461575751341568X^{11}Y^{5}Z^{3}+295405392629564000X^{10}Y^{6}Z^{3}+295858302215332416X^{9}Y^{7}Z^{3}+205184420845854128X^{8}Y^{8}Z^{3}+68998839526617728X^{7}Y^{9}Z^{3}-37895823180510960X^{6}Y^{10}Z^{3}-74431187753605760X^{5}Y^{11}Z^{3}-57514051352046240X^{4}Y^{12}Z^{3}-27511500975153216X^{3}Y^{13}Z^{3}-8325048434781440X^{2}Y^{14}Z^{3}-1467919037548800XY^{15}Z^{3}-110123745097984Y^{16}Z^{3}+27174758023168X^{15}Z^{4}+167949011112960X^{14}YZ^{4}+320748234376192X^{13}Y^{2}Z^{4}-552484064676992X^{12}Y^{3}Z^{4}-4928274553238848X^{11}Y^{4}Z^{4}-15195312488459180X^{10}Y^{5}Z^{4}-29634134207295520X^{9}Y^{6}Z^{4}-40517362495561980X^{8}Y^{7}Z^{4}-39774056928479216X^{7}Y^{8}Z^{4}-27511541290987859X^{6}Y^{9}Z^{4}-12438027121564408X^{5}Y^{10}Z^{4}-2795664941321000X^{4}Y^{11}Z^{4}+356546605693992X^{3}Y^{12}Z^{4}+425585543370912X^{2}Y^{13}Z^{4}+103344035107456XY^{14}Z^{4}+12469923411648Y^{15}Z^{4}-5180561713152X^{14}Z^{5}-45436834401280X^{13}YZ^{5}-215546409297408X^{12}Y^{2}Z^{5}-692714527147328X^{11}Y^{3}Z^{5}-1632000337040248X^{10}Y^{4}Z^{5}-2918424308174144X^{9}Y^{5}Z^{5}-4044213270045788X^{8}Y^{6}Z^{5}-4390280164844224X^{7}Y^{7}Z^{5}-3811060424286906X^{6}Y^{8}Z^{5}-2740641172131716X^{5}Y^{9}Z^{5}-1654444906677101X^{4}Y^{10}Z^{5}-793688449617742X^{3}Y^{11}Z^{5}-262378225299656X^{2}Y^{12}Z^{5}-43307996591992XY^{13}Z^{5}-273222171024Y^{14}Z^{5}-738496890880X^{13}Z^{6}-4853095738624X^{12}YZ^{6}-14716366244608X^{11}Y^{2}Z^{6}-19939890145616X^{10}Y^{3}Z^{6}+22196090368704X^{9}Y^{4}Z^{6}+175641732165956X^{8}Y^{5}Z^{6}+440052554351872X^{7}Y^{6}Z^{6}+682234184879212X^{6}Y^{7}Z^{6}+741224238861512X^{5}Y^{8}Z^{6}+588189966228347X^{4}Y^{9}Z^{6}+339099777645552X^{3}Y^{10}Z^{6}+134774132250374X^{2}Y^{11}Z^{6}+32374839534246XY^{12}Z^{6}+3330366959352Y^{13}Z^{6}+59935106560X^{12}Z^{7}+504414373120X^{11}YZ^{7}+2008509183072X^{10}Y^{2}Z^{7}+4260976408192X^{9}Y^{3}Z^{7}+3738590127496X^{8}Y^{4}Z^{7}-5084300302536X^{7}Y^{5}Z^{7}-22035626741408X^{6}Y^{6}Z^{7}-38442320798056X^{5}Y^{7}Z^{7}-44248215037922X^{4}Y^{8}Z^{7}-35476294464590X^{3}Y^{9}Z^{7}-18931281839512X^{2}Y^{10}Z^{7}-5942055675032XY^{11}Z^{7}-803226242631Y^{12}Z^{7}+5903156224X^{11}Z^{8}+37415340032X^{10}YZ^{8}+193623093888X^{9}Y^{2}Z^{8}+720337998912X^{8}Y^{3}Z^{8}+1795466779232X^{7}Y^{4}Z^{8}+3019884731752X^{6}Y^{5}Z^{8}+3606775502680X^{5}Y^{6}Z^{8}+3459914020976X^{4}Y^{7}Z^{8}+2835079713256X^{3}Y^{8}Z^{8}+1776869417869X^{2}Y^{9}Z^{8}+686553842226XY^{10}Z^{8}+113118568289Y^{11}Z^{8}-456074752X^{10}Z^{9}-5072755968X^{9}YZ^{9}-27622025984X^{8}Y^{2}Z^{9}-89894098560X^{7}Y^{3}Z^{9}-190503539072X^{6}Y^{4}Z^{9}-264642989072X^{5}Y^{5}Z^{9}-266146725768X^{4}Y^{6}Z^{9}-229491020976X^{3}Y^{7}Z^{9}-167516198528X^{2}Y^{8}Z^{9}-79350158888XY^{9}Z^{9}-16148212644Y^{10}Z^{9}+162883840X^{8}YZ^{10}+816263168X^{7}Y^{2}Z^{10}+2353040368X^{6}Y^{3}Z^{10}+3373363840X^{5}Y^{4}Z^{10}+2092366836X^{4}Y^{5}Z^{10}+2151155272X^{3}Y^{6}Z^{10}+4935238308X^{2}Y^{7}Z^{10}+4483426864XY^{8}Z^{10}+1297938096Y^{9}Z^{10}+1745184X^{6}Y^{2}Z^{11}+51609152X^{5}Y^{3}Z^{11}+241842440X^{4}Y^{4}Z^{11}+345844744X^{3}Y^{5}Z^{11}+63424228X^{2}Y^{6}Z^{11}-169606248XY^{7}Z^{11}-79856280Y^{8}Z^{11}-2908640X^{4}Y^{3}Z^{12}-6552672X^{3}Y^{4}Z^{12}+1559964X^{2}Y^{5}Z^{12}+10270792XY^{6}Z^{12}+5004468Y^{7}Z^{12}-145432X^{2}Y^{4}Z^{13}-290864XY^{5}Z^{13}-145432Y^{6}Z^{13} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
112.192.5-56.bl.1.9 $112$ $2$ $2$ $5$ $?$
112.192.5-56.bl.1.31 $112$ $2$ $2$ $5$ $?$