| $\GL_2(\Z/44\Z)$-generators: |
$\begin{bmatrix}5&22\\37&37\end{bmatrix}$, $\begin{bmatrix}31&11\\1&8\end{bmatrix}$, $\begin{bmatrix}38&11\\15&26\end{bmatrix}$, $\begin{bmatrix}43&33\\3&32\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
44.240.6-44.b.1.1, 44.240.6-44.b.1.2, 44.240.6-44.b.1.3, 44.240.6-44.b.1.4, 44.240.6-44.b.1.5, 44.240.6-44.b.1.6, 88.240.6-44.b.1.1, 88.240.6-44.b.1.2, 88.240.6-44.b.1.3, 88.240.6-44.b.1.4, 88.240.6-44.b.1.5, 88.240.6-44.b.1.6, 132.240.6-44.b.1.1, 132.240.6-44.b.1.2, 132.240.6-44.b.1.3, 132.240.6-44.b.1.4, 132.240.6-44.b.1.5, 132.240.6-44.b.1.6, 220.240.6-44.b.1.1, 220.240.6-44.b.1.2, 220.240.6-44.b.1.3, 220.240.6-44.b.1.4, 220.240.6-44.b.1.5, 220.240.6-44.b.1.6, 264.240.6-44.b.1.1, 264.240.6-44.b.1.2, 264.240.6-44.b.1.3, 264.240.6-44.b.1.4, 264.240.6-44.b.1.5, 264.240.6-44.b.1.6, 308.240.6-44.b.1.1, 308.240.6-44.b.1.2, 308.240.6-44.b.1.3, 308.240.6-44.b.1.4, 308.240.6-44.b.1.5, 308.240.6-44.b.1.6 |
| Cyclic 44-isogeny field degree: |
$6$ |
| Cyclic 44-torsion field degree: |
$120$ |
| Full 44-torsion field degree: |
$10560$ |
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 $ |
| $=$ | $y^{2} - 2 y z - y w - z^{2} + 3 z t + 2 z u - w t - t^{2} + t u + u^{2}$ |
| $=$ | $y^{2} + 2 y w + y t + y u + 2 z^{2} - 2 z w - z t - z u - w t - 3 w u + t u$ |
| $=$ | $y z - 2 y w - y t - y u + z^{2} - 2 z w - z t - z u + w^{2} + 3 w t - t u - u^{2}$ |
| $=$ | $\cdots$ |
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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} $ |
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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 between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y-w+\frac{1}{3}u$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{11}{3}x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}z-\frac{2}{3}w+\frac{2}{3}t+\frac{1}{3}u$ |
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}}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
