↑

Properties

Label 20.288.3-20.b.2.1
Level $20$
Index $288$
Genus $3$
Analytic rank $0$
Cusps $20$
$\Q$-cusps $2$

Related objects

Downloads

Learn more

Invariants

Level: $20$ $\SL_2$-level: $20$ Newform level: $100$
Index: $288$ $\PSL_2$-index:$144$
Genus: $3 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$
Cusps: $20$ (of which $2$ are rational) Cusp widths $2^{8}\cdot4^{2}\cdot10^{8}\cdot20^{2}$ Cusp orbits $1^{2}\cdot2^{5}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 3$
$\overline{\Q}$-gonality: $2 \le \gamma \le 3$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20R3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 20.288.3.2

Level structure

$\GL_2(\Z/20\Z)$-generators: $\begin{bmatrix}3&4\\2&15\end{bmatrix}$, $\begin{bmatrix}13&14\\4&9\end{bmatrix}$, $\begin{bmatrix}15&8\\16&3\end{bmatrix}$, $\begin{bmatrix}17&2\\0&11\end{bmatrix}$
$\GL_2(\Z/20\Z)$-subgroup: $C_2^3\times F_5$
Contains $-I$: no $\quad$ (see 20.144.3.b.2 for the level structure with $-I$)
Cyclic 20-isogeny field degree: $2$
Cyclic 20-torsion field degree: $4$
Full 20-torsion field degree: $160$

Jacobian

Conductor: $2^{6}\cdot5^{5}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 20.2.a.a, 100.2.e.b

Models

Canonical model in $\mathbb{P}^{ 2 }$

$ 0 $ $=$ $ x^{3} y - 2 x^{3} z + x^{2} y^{2} - x^{2} y z - 2 x^{2} z^{2} + 2 x y^{2} z - 5 x y z^{2} + 2 x z^{3} + \cdots + y z^{3} $
 Copy content Toggle raw display

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:1)$, $(1:0:0)$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{5^4}\cdot\frac{4096x^{36}+147456x^{35}z+2580480x^{34}z^{2}+29245440x^{33}z^{3}+236359680x^{32}z^{4}+1386872832x^{31}z^{5}+5540216832x^{30}z^{6}+10304225280x^{29}z^{7}-36031119360x^{28}z^{8}-334102200320x^{27}z^{9}-732338356224x^{26}z^{10}+3607093641216x^{25}z^{11}+28711189299200x^{24}z^{12}+26676550041600x^{23}z^{13}-563694349516800x^{22}z^{14}-2692448147210240x^{21}z^{15}+4063854137794560x^{20}z^{16}+82641632774553600x^{19}z^{17}+195287542232268800x^{18}z^{18}-1449619869152870400x^{17}z^{19}-10449655141768765440x^{16}z^{20}+1143396326353141760x^{15}z^{21}+287164708455535411200x^{14}z^{22}+1014675315176118681600x^{13}z^{23}-4299721756433224908800x^{12}z^{24}-43617708073224481603584x^{11}z^{25}-39302376469975659085824x^{10}z^{26}+1102651778911245038714880x^{9}z^{27}+5025102488005524071792640x^{8}z^{28}-12602130759456579500113920x^{7}z^{29}-194722193228331136952762368x^{6}z^{30}-310105991454249748374355968x^{5}z^{31}+4271637858530930746990202880x^{4}z^{32}+26259250339574711503974973440x^{3}z^{33}-3300101336248x^{2}y^{34}+4971822398464x^{2}y^{33}z-90354138207312x^{2}y^{32}z^{2}+13386297410656x^{2}y^{31}z^{3}-1053521175487568x^{2}y^{30}z^{4}-2375811739944384x^{2}y^{29}z^{5}-9879681635303488x^{2}y^{28}z^{6}-45871779932708096x^{2}y^{27}z^{7}-149694564438236352x^{2}y^{26}z^{8}-535511481652306944x^{2}y^{25}z^{9}-2093764544771809280x^{2}y^{24}z^{10}-6788407350380328960x^{2}y^{23}z^{11}-22688781081880632320x^{2}y^{22}z^{12}-80127576781679779840x^{2}y^{21}z^{13}-249863189862920724480x^{2}y^{20}z^{14}-775114009864275230720x^{2}y^{19}z^{15}-2530828425906409943040x^{2}y^{18}z^{16}-7430511747449482567680x^{2}y^{17}z^{17}-21163539446784111452160x^{2}y^{16}z^{18}-64377785793175957995520x^{2}y^{15}z^{19}-176210758407501587742720x^{2}y^{14}z^{20}-452286392981956797399040x^{2}y^{13}z^{21}-1295959315965427622215680x^{2}y^{12}z^{22}-3301869250290206838620160x^{2}y^{11}z^{23}-7258785300910464229867520x^{2}y^{10}z^{24}-20015249251483700421853184x^{2}y^{9}z^{25}-48128399774839410255986688x^{2}y^{8}z^{26}-73340092342147303754039296x^{2}y^{7}z^{27}-209623863468034181675876352x^{2}y^{6}z^{28}-528540316872593651292831744x^{2}y^{5}z^{29}+167183932421828834477211648x^{2}y^{4}z^{30}-427077873379544191141412864x^{2}y^{3}z^{31}-3923370704756507931168473088x^{2}y^{2}z^{32}+25960427398360330782487609344x^{2}yz^{33}+17615684356241310760876597248x^{2}z^{34}+3008846030520xy^{35}+552447037104xy^{34}z+102182877394528xy^{33}z^{2}+39392538858176xy^{32}z^{3}+1136769954629712xy^{31}z^{4}+1043778707953984xy^{30}z^{5}+1843783579996032xy^{29}z^{6}-5657018755810176xy^{28}z^{7}-96647528467049792xy^{27}z^{8}-509811333795829504xy^{26}z^{9}-2169216895074731008xy^{25}z^{10}-9298743995221442560xy^{24}z^{11}-35366887880400081920xy^{23}z^{12}-123974346241482096640xy^{22}z^{13}-440973417822228295680xy^{21}z^{14}-1490601561661931970560xy^{20}z^{15}-4758121854927668797440xy^{19}z^{16}-15342830905874967982080xy^{18}z^{17}-47935445342157616783360xy^{17}z^{18}-141541389590547132088320xy^{16}z^{19}-421046333002853598883840xy^{15}z^{20}-1226830569891155038781440xy^{14}z^{21}-3341507386731642807214080xy^{13}z^{22}-9169087367812918729359360xy^{12}z^{23}-25013085910076902837944320xy^{11}z^{24}-61885332441937354771988480xy^{10}z^{25}-154065485010350192295149568xy^{9}z^{26}-394480571581985321346072576xy^{8}z^{27}-842713939677227242523656192xy^{7}z^{28}-1751350266319144054653845504xy^{6}z^{29}-4179098702623125372691939328xy^{5}z^{30}-6233389687727663392395362304xy^{4}z^{31}-1126478473021082110109155328xy^{3}z^{32}-4907165255818700186544766976xy^{2}z^{33}+45757712988528866594587148288xyz^{34}-21862321107007118717877075968xz^{35}-5107586099303y^{36}-4247330739064y^{35}z-185767993680408y^{34}z^{2}-505195570438576y^{33}z^{3}-3508435730399552y^{32}z^{4}-14372426849247712y^{31}z^{5}-55183963634249312y^{30}z^{6}-221306168358920064y^{29}z^{7}-769175660718053808y^{28}z^{8}-2548815185448716416y^{27}z^{9}-8456684524679267584y^{26}z^{10}-25897562541164400640y^{25}z^{11}-75927461947782842880y^{24}z^{12}-225440795805198223360y^{23}z^{13}-632659212480769054720y^{22}z^{14}-1711564011848129945600y^{21}z^{15}-4842519673694223464960y^{20}z^{16}-13241476522271931269120y^{19}z^{17}-35043776882097057648640y^{18}z^{18}-101211066833346972897280y^{17}z^{19}-286864542827936179978240y^{16}z^{20}-768664853732848124968960y^{15}z^{21}-2291898163652332146053120y^{14}z^{22}-6696312090937881020784640y^{13}z^{23}-17463554283714196675297280y^{12}z^{24}-50837077005130857723346944y^{11}z^{25}-146952161201007664000090112y^{10}z^{26}-346643130412842357546942464y^{9}z^{27}-922662284227857692280619008y^{8}z^{28}-2639759573876419698614665216y^{7}z^{29}-4868292818237423575992958976y^{6}z^{30}-10287511544350519457947713536y^{5}z^{31}-34009250656178220240617275392y^{4}z^{32}-5661283185052593850901069824y^{3}z^{33}+18474935405204105607075069952y^{2}z^{34}-10931160553503559358938611712yz^{35}+4096z^{36}}{64000000x^{16}z^{20}+1024000000x^{15}z^{21}+7680000000x^{14}z^{22}+3840000000x^{13}z^{23}-344320000000x^{12}z^{24}-2754048000000x^{11}z^{25}-796288000000x^{10}z^{26}+101276160000000x^{9}z^{27}+662743680000000x^{8}z^{28}-889501440000000x^{7}z^{29}-26250994688000000x^{6}z^{30}-120714946048000000x^{5}z^{31}+541552540480000000x^{4}z^{32}+5645363107840000000x^{3}z^{33}-14580x^{2}y^{34}+1027890x^{2}y^{33}z+4592700x^{2}y^{32}z^{2}+1414800x^{2}y^{31}z^{3}+7056450x^{2}y^{30}z^{4}+70567470x^{2}y^{29}z^{5}+63396400x^{2}y^{28}z^{6}+105144200x^{2}y^{27}z^{7}+645290750x^{2}y^{26}z^{8}+875715250x^{2}y^{25}z^{9}+1763229200x^{2}y^{24}z^{10}+5439419700x^{2}y^{23}z^{11}+9567370150x^{2}y^{22}z^{12}+21048125000x^{2}y^{21}z^{13}+48319669000x^{2}y^{20}z^{14}+97387622400x^{2}y^{19}z^{15}+210941542000x^{2}y^{18}z^{16}+448289684800x^{2}y^{17}z^{17}+939170708000x^{2}y^{16}z^{18}+1985983224000x^{2}y^{15}z^{19}+3962395506400x^{2}y^{14}z^{20}+8943113129600x^{2}y^{13}z^{21}+15701869513600x^{2}y^{12}z^{22}+22754530880000x^{2}y^{11}z^{23}+96777174672000x^{2}y^{10}z^{24}+6274610472960x^{2}y^{9}z^{25}-278381368519680x^{2}y^{8}z^{26}+2633710156748800x^{2}y^{7}z^{27}-3074100021017600x^{2}y^{6}z^{28}-13975790218342400x^{2}y^{5}z^{29}+140040503416156160x^{2}y^{4}z^{30}-125527016448000000x^{2}y^{3}z^{31}-474525935776000000x^{2}y^{2}z^{32}+5564202331008000000x^{2}yz^{33}+4317668638336000000x^{2}z^{34}+136809xy^{35}+2401407xy^{34}z+5537295xy^{33}z^{2}-7520940xy^{32}z^{3}-2071815xy^{31}z^{4}+33133457xy^{30}z^{5}-214621379xy^{29}z^{6}-610803000xy^{28}z^{7}-1399434725xy^{27}z^{8}-5869599225xy^{26}z^{9}-16482105135xy^{25}z^{10}-44090331030xy^{24}z^{11}-125450772300xy^{23}z^{12}-330214499100xy^{22}z^{13}-856645969600xy^{21}z^{14}-2202859434640xy^{20}z^{15}-5539041408720xy^{19}z^{16}-13772715932800xy^{18}z^{17}-33847303776800xy^{17}z^{18}-82266373460800xy^{16}z^{19}-197952715764640xy^{15}z^{20}-473263319320320xy^{14}z^{21}-1118268842188800xy^{13}z^{22}-2622648326867200xy^{12}z^{23}-6163545814851200xy^{11}z^{24}-14066404536355328xy^{10}z^{25}-32008519326930944xy^{9}z^{26}-74404274402058240xy^{8}z^{27}-154035202414289920xy^{7}z^{28}-318413796244561920xy^{6}z^{29}-719713594084347904xy^{5}z^{30}-915688567228096512xy^{4}z^{31}-253657522656000000xy^{3}z^{32}-523446130432000000xy^{2}z^{33}+10084333888896000000xyz^{34}-4932642265088000000xz^{35}+305937y^{36}+1099656y^{35}z-6759315y^{34}z^{2}-33708645y^{33}z^{3}-71075895y^{32}z^{4}-283633264y^{31}z^{5}-1011809727y^{30}z^{6}-2602019175y^{29}z^{7}-7396456800y^{28}z^{8}-20818213300y^{27}z^{9}-53570888805y^{26}z^{10}-138774766065y^{25}z^{11}-353798905125y^{24}z^{12}-878695669050y^{23}z^{13}-2163633971800y^{22}z^{14}-5260279855920y^{21}z^{15}-12639673094735y^{20}z^{16}-30102616302600y^{19}z^{17}-71064131963400y^{18}z^{18}-166465422298400y^{17}z^{19}-387556013111920y^{16}z^{20}-895274947026560y^{15}z^{21}-2061257630356800y^{14}z^{22}-4737485212633600y^{13}z^{23}-10747222333997600y^{12}z^{24}-24653025044418304y^{11}z^{25}-56940361382961152y^{10}z^{26}-125668059545605120y^{9}z^{27}-292361386075791360y^{8}z^{28}-696582413047695360y^{7}z^{29}-1336006207904212992y^{6}z^{30}-2980700726366048256y^{5}z^{31}-7749297850364000000y^{4}z^{32}-947103259488000000y^{3}z^{33}+3962749784864000000y^{2}z^{34}-2466321132544000000yz^{35}}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_1(2,10)$ $10$ $2$ $2$ $1$ $0$ $2$
20.144.1-10.a.1.4 $20$ $2$ $2$ $1$ $0$ $2$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
20.576.9-20.c.2.1 $20$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
20.576.9-20.d.2.1 $20$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
20.576.9-20.d.3.4 $20$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
20.576.9-20.e.1.2 $20$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
20.576.9-20.e.3.1 $20$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
20.576.9-20.f.1.2 $20$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
20.576.9-20.f.3.4 $20$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
20.576.13-20.k.2.8 $20$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{2}\cdot4$
20.576.13-20.l.2.8 $20$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{2}\cdot4$
20.576.13-20.q.1.8 $20$ $2$ $2$ $13$ $1$ $1^{2}\cdot2^{2}\cdot4$
20.576.13-20.r.2.8 $20$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{2}\cdot4$
20.1440.31-20.b.1.7 $20$ $5$ $5$ $31$ $0$ $1^{6}\cdot2^{7}\cdot8$
40.576.9-40.f.2.2 $40$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
40.576.9-40.f.3.3 $40$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
40.576.9-40.h.2.2 $40$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
40.576.9-40.h.3.3 $40$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
40.576.9-40.k.2.2 $40$ $2$ $2$ $9$ $2$ $1^{2}\cdot2^{2}$
40.576.9-40.k.3.3 $40$ $2$ $2$ $9$ $2$ $1^{2}\cdot2^{2}$
40.576.9-40.n.2.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
40.576.9-40.n.3.3 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
40.576.13-40.ca.2.15 $40$ $2$ $2$ $13$ $1$ $1^{2}\cdot2^{2}\cdot4$
40.576.13-40.cd.2.15 $40$ $2$ $2$ $13$ $1$ $1^{2}\cdot2^{2}\cdot4$
40.576.13-40.cs.2.15 $40$ $2$ $2$ $13$ $2$ $1^{2}\cdot2^{2}\cdot4$
40.576.13-40.cv.2.15 $40$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{2}\cdot4$
60.576.9-60.o.2.2 $60$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
60.576.9-60.o.3.4 $60$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
60.576.9-60.p.2.2 $60$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
60.576.9-60.p.3.3 $60$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
60.576.9-60.u.2.2 $60$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
60.576.9-60.u.3.4 $60$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
60.576.9-60.v.2.2 $60$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
60.576.9-60.v.4.2 $60$ $2$ $2$ $9$ $1$ $1^{2}\cdot2^{2}$
60.576.13-60.eo.2.15 $60$ $2$ $2$ $13$ $0$ $1^{2}\cdot2^{2}\cdot4$
60.576.13-60.ep.1.15 $60$ $2$ $2$ $13$ $1$ $1^{2}\cdot2^{2}\cdot4$
60.576.13-60.fg.2.15 $60$ $2$ $2$ $13$ $1$ $1^{2}\cdot2^{2}\cdot4$
60.576.13-60.fh.1.14 $60$ $2$ $2$ $13$ $1$ $1^{2}\cdot2^{2}\cdot4$
60.864.27-60.h.1.2 $60$ $3$ $3$ $27$ $2$ $1^{6}\cdot2^{5}\cdot8$
60.1152.29-60.l.1.4 $60$ $4$ $4$ $29$ $0$ $1^{6}\cdot2^{4}\cdot12$