Modular curve 88.240.6-44.b.1.2

• Label: 88.240.6-44.b.1.2
• Level: $88$
• Index: $240$
• Genus: $6$
• Cusps: $10$
• $\Q$-cusps: $0$

Invariants

Level:	$88$		$\SL_2$-level:	$22$		Newform level:	$1936$
Index:	$240$		$\PSL_2$-index:	$120$
Genus:	$6 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps:	$10$ (none of which are rational)		Cusp widths	$2^{5}\cdot22^{5}$		Cusp orbits	$5^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 10$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 6$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

22A6

Level structure

$\GL_2(\Z/88\Z)$-generators:	$\begin{bmatrix}18&53\\33&31\end{bmatrix}$, $\begin{bmatrix}67&19\\70&19\end{bmatrix}$, $\begin{bmatrix}67&54\\61&19\end{bmatrix}$, $\begin{bmatrix}85&83\\40&31\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 44.120.6.b.1 for the level structure with $-I$)
Cyclic 88-isogeny field degree:	$12$
Cyclic 88-torsion field degree:	$480$
Full 88-torsion field degree:	$84480$

Models

Canonical model in $\mathbb{P}^{ 5 }$ defined by 6 equations

$ 0 $	$=$	$ 3 y^{2} + 2 y z - 2 y t + z^{2} - z w - 2 z t - w u + t^{2} + t u $
	$=$	$y^{2} - 2 y z - y w - z^{2} + 3 z t + 2 z u - w t - t^{2} + t u + u^{2}$
	$=$	$y^{2} + 2 y w + y t + y u + 2 z^{2} - 2 z w - z t - z u - w t - 3 w u + t u$
	$=$	$y z - 2 y w - y t - y u + z^{2} - 2 z w - z t - z u + w^{2} + 3 w t - t u - u^{2}$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{10} - 8 x^{9} z - 2 x^{8} y^{2} + 9 x^{8} z^{2} + 14 x^{7} y^{2} z + 60 x^{7} z^{3} + x^{6} y^{4} + \cdots + z^{10} $

Rational points

This modular curve has 1 known rational point but no rational cusps or CM points. The following are the known rational points on this modular curve (one row per $j$-invariant).

Elliptic curve	CM	$j$-invariant		$j$-height	Canonical model	Plane model
121.a1	no	$-121$	$= -1 \cdot 11^{2}$	$4.796$	$(-1:-1:5:3:0:1)$, $(-1:1:0:5:-1:1)$, $(-1/3:-1/3:-1/3:0:-1/3:1)$, $(-1/5:1/5:-1/5:-1/5:-3/5:1)$, $(1/5:1/5:-1/5:-1/5:-3/5:1)$, $(1/3:-1/3:-1/3:0:-1/3:1)$, $(1:-1:5:3:0:1)$, $(1:1:0:5:-1:1)$, $(-1/5:1/5:3/5:1/5:1:0)$, $(1/5:1/5:3/5:1/5:1:0)$	$(1:1:0)$, $(1:1:1)$, $(0:1:0)$, $(-1:1:0)$, $(1:-1:1)$, $(0:-1:1)$, $(0:1:1)$

Maps to other modular curves

$j$-invariant map of degree 120 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{22798100008652523009276933yt^{11}+113685658228800155180180184yt^{10}u+215711266798608656505878534yt^{9}u^{2}+199513958940240237219563716yt^{8}u^{3}+104657971186548415096322384yt^{7}u^{4}+40941182637395609936680752yt^{6}u^{5}+10310169752575001439797307yt^{5}u^{6}-442066927698600730518766yt^{4}u^{7}+134320896482502100807216yt^{3}u^{8}+486633393547231649610434yt^{2}u^{9}+129657523272990322386444ytu^{10}+9855357895926895789998yu^{11}+9896522320290043412874825zt^{11}+51438500128297712330743145zt^{10}u+103046314426693457258421884zt^{9}u^{2}+101913811068797424438477466zt^{8}u^{3}+56618606943428534624873494zt^{7}u^{4}+22189604594997230450492316zt^{6}u^{5}+5895013187621069843069099zt^{5}u^{6}-239000054725149057731641zt^{4}u^{7}-131708488414041134431784zt^{3}u^{8}+169273410904440745542994zt^{2}u^{9}+54225419625513516304866ztu^{10}+4381082776551462988104zu^{11}-4653568671289632402819909w^{2}t^{10}-19995633058271750404669490w^{2}t^{9}u-30145146192451028944816300w^{2}t^{8}u^{2}-19390694933103825375238040w^{2}t^{7}u^{3}-6932191819663157137919700w^{2}t^{6}u^{4}-2646071678428608358110738w^{2}t^{5}u^{5}+191956942803456954028435w^{2}t^{4}u^{6}+163988128119982992318800w^{2}t^{3}u^{7}-89416873724619400614890w^{2}t^{2}u^{8}-35431609856361376637250w^{2}tu^{9}-3089541844049550931854w^{2}u^{10}-23505631503748441329113274wt^{11}-100557352195120699058864731wt^{10}u-150472674313670450492715902wt^{9}u^{2}-96410639472468788695660468wt^{8}u^{3}-36292298254114200541972842wt^{7}u^{4}-15533883380670192776272686wt^{6}u^{5}-160290328221716087451664wt^{5}u^{6}+181491905977203310347823wt^{4}u^{7}-600002687676821788981768wt^{3}u^{8}-157952462547231931119192wt^{2}u^{9}-4488556261559189736612wtu^{10}+990126865938672109692wu^{11}-5610264346347294604281075t^{12}-27088412269260149556353327t^{11}u-45036377185830463382811403t^{10}u^{2}-24714301042045007150716964t^{9}u^{3}+7312762012091159492274234t^{8}u^{4}+11986658985625714384630836t^{7}u^{5}+6520769280831680427664705t^{6}u^{6}+3427816534748668860448385t^{5}u^{7}+138414372140845011279361t^{4}u^{8}-217578373623851713199766t^{3}u^{9}+23050187777577232605336t^{2}u^{10}+20567102205796285791582tu^{11}+2034093176778878375838u^{12}}{49193909027010173202yt^{11}-21582033107566626644yt^{10}u-66025400659625983559yt^{9}u^{2}-73745290908374883661yt^{8}u^{3}+72342957402766188406yt^{7}u^{4}+7058371574463471683yt^{6}u^{5}-32537863988850336972yt^{5}u^{6}+234622735820309258841yt^{4}u^{7}-203269175701337146511yt^{3}u^{8}+64962921836172725331yt^{2}u^{9}+54535721445711789006ytu^{10}-92226505637904080728yu^{11}+20882063711455927300zt^{11}-4581736685327615370zt^{10}u-31833027598081289129zt^{9}u^{2}-35483997468780184966zt^{8}u^{3}+25017010055302401601zt^{7}u^{4}+11054080427750907469zt^{6}u^{5}-17239499630756896839zt^{5}u^{6}+102364987239722700121zt^{4}u^{7}-69148340721845044816zt^{3}u^{8}-2793667706442668424zt^{2}u^{9}+46710528462024184319ztu^{10}-43379859361776893894zu^{11}-8669071486874269696w^{2}t^{10}+8942953787740234310w^{2}t^{9}u+5533161673951143355w^{2}t^{8}u^{2}+11045119225535538935w^{2}t^{7}u^{3}-20835113694098749520w^{2}t^{6}u^{4}+14015867841276973173w^{2}t^{5}u^{5}-2136155635826671740w^{2}t^{4}u^{6}-46656313419637020745w^{2}t^{3}u^{7}+72797183027431996035w^{2}t^{2}u^{8}-62096822617066177095w^{2}tu^{9}+29089985465823962644w^{2}u^{10}-51341074229374184756wt^{11}+58364311989155519356wt^{10}u+28997290436155156322wt^{9}u^{2}+55640752140443881783wt^{8}u^{3}-113150425187478065983wt^{7}u^{4}+70023514944569730036wt^{6}u^{5}-15343570184289335331wt^{5}u^{6}-234435519748206190328wt^{4}u^{7}+377937401958163828383wt^{3}u^{8}-327434920836426185433wt^{2}u^{9}+162575241647880293417wtu^{10}-9383909055363538362wu^{11}-11753041962979364800t^{12}+7512944076615217262t^{11}u+22278904514404879603t^{10}u^{2}+6514364672074869614t^{9}u^{3}-25805504001343010264t^{8}u^{4}-6942779333305953726t^{7}u^{5}+25041735493444541025t^{6}u^{6}-66892125748421583340t^{5}u^{7}+60396852308103316314t^{4}u^{8}+11080060295468687211t^{3}u^{9}-65871933261688885076t^{2}u^{10}+67605688603114948563tu^{11}-20131244780793878718u^{12}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 44.120.6.b.1 :

$\displaystyle X$	$=$	$\displaystyle y-w+\frac{1}{3}u$
$\displaystyle Y$	$=$	$\displaystyle \frac{11}{3}x$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{3}z-\frac{2}{3}w+\frac{2}{3}t+\frac{1}{3}u$

Equation of the image curve:

$0$

$=$

$ X^{10}-2X^{8}Y^{2}+X^{6}Y^{4}-8X^{9}Z+14X^{7}Y^{2}Z-6X^{5}Y^{4}Z+9X^{8}Z^{2}-59X^{6}Y^{2}Z^{2}+15X^{4}Y^{4}Z^{2}+60X^{7}Z^{3}+135X^{5}Y^{2}Z^{3}-20X^{3}Y^{4}Z^{3}-161X^{6}Z^{4}-142X^{4}Y^{2}Z^{4}+15X^{2}Y^{4}Z^{4}+67X^{5}Z^{5}+52X^{3}Y^{2}Z^{5}-6XY^{4}Z^{5}+135X^{4}Z^{6}+3X^{2}Y^{2}Z^{6}+Y^{4}Z^{6}-112X^{3}Z^{7}-6X^{2}Z^{8}-2Y^{2}Z^{8}+15XZ^{9}+Z^{10} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
88.48.2-44.a.1.2	$88$	$5$	$5$	$2$	$?$
88.120.1-11.b.1.2	$88$	$2$	$2$	$1$	$?$
88.120.1-11.b.1.8	$88$	$2$	$2$	$1$	$?$

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

Covering curve	Level	Index	Degree	Genus
88.480.16-44.b.2.2	$88$	$2$	$2$	$16$
88.480.16-88.b.2.5	$88$	$2$	$2$	$16$
88.480.16-44.h.1.2	$88$	$2$	$2$	$16$
88.480.16-88.h.1.3	$88$	$2$	$2$	$16$
264.480.16-132.b.2.4	$264$	$2$	$2$	$16$
264.480.16-264.b.1.1	$264$	$2$	$2$	$16$
264.480.16-132.h.2.4	$264$	$2$	$2$	$16$
264.480.16-264.h.2.1	$264$	$2$	$2$	$16$