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Properties

Label 38.240.4-38.a.1.1
Level $38$
Index $240$
Genus $4$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $3$

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Invariants

Level: $38$ $\SL_2$-level: $38$ Newform level: $76$
Index: $240$ $\PSL_2$-index:$120$
Genus: $4 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 12 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $3$ are rational) Cusp widths $2^{3}\cdot38^{3}$ Cusp orbits $1^{3}\cdot3$
Elliptic points: $0$ of order $2$ and $12$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3$
Rational cusps: $3$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 38B4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 38.240.4.10

Level structure

$\GL_2(\Z/38\Z)$-generators: $\begin{bmatrix}4&25\\9&25\end{bmatrix}$, $\begin{bmatrix}16&29\\35&37\end{bmatrix}$, $\begin{bmatrix}17&22\\12&27\end{bmatrix}$
Contains $-I$: no $\quad$ (see 38.120.4.a.1 for the level structure with $-I$)
Cyclic 38-isogeny field degree: $3$
Cyclic 38-torsion field degree: $54$
Full 38-torsion field degree: $3078$

Jacobian

Conductor: $2^{6}\cdot19^{4}$
Simple: no
Squarefree: yes
Decomposition: $1^{2}\cdot2$
Newforms: 19.2.a.a, 76.2.a.a, 76.2.e.a

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 19 x^{2} - 6 y^{2} + y z - y w + z^{2} - z w + w^{2} $
$=$ $18 y^{3} + 18 y^{2} z - 11 y^{2} w + 25 y z^{2} - 20 y z w + 4 y w^{2} + 7 z^{3} - 16 z^{2} w + \cdots + 7 w^{3}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 1026 x^{6} - 684 x^{4} y^{2} + 399 x^{4} y z - 152 x^{4} z^{2} - 189 x^{2} y^{4} + 185 x^{2} y^{3} z + \cdots + 14 y z^{5} $
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Rational points

This modular curve has 3 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:-1/2:0:1)$, $(0:-1/3:1:0)$, $(0:1/3:2/3:1)$

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{1}{2\cdot3}\cdot\frac{1883174313020502334031110656y^{2}z^{18}-4005401209769664132928846848y^{2}z^{17}w-6341309448228633360845719296y^{2}z^{16}w^{2}+1290961706421336697814999040y^{2}z^{15}w^{3}-10902079138839596514295204608y^{2}z^{14}w^{4}+57871566724598560295554715136y^{2}z^{13}w^{5}-12216621186358589490940876032y^{2}z^{12}w^{6}-92531823756796873776902819328y^{2}z^{11}w^{7}+140248871189835525136823569632y^{2}z^{10}w^{8}-83966297417238697942907983680y^{2}z^{9}w^{9}-53566043635226740234152575664y^{2}z^{8}w^{10}+141807417234255187067704589568y^{2}z^{7}w^{11}-114597498673478566211753556360y^{2}z^{6}w^{12}+42864870329214065921619646992y^{2}z^{5}w^{13}-413920570228423616588549472y^{2}z^{4}w^{14}-6904837197210752239573408464y^{2}z^{3}w^{15}+2998742890159795691356504002y^{2}z^{2}w^{16}-559680132434438146941536508y^{2}zw^{17}+39907810973040426232753037y^{2}w^{18}+1270494616552397719330323456yz^{19}-8357960266212269439041160960yz^{18}w-5235838389133776200557416960yz^{17}w^{2}+38659104909441699803016786432yz^{16}w^{3}-21349479358005851992209283584yz^{15}w^{4}+9925873097256210960670029696yz^{14}w^{5}+45874633129852065195913122816yz^{13}w^{6}-183380186570430964079335216896yz^{12}w^{7}+165964079425429583487405995328yz^{11}w^{8}+46966056165334262608299342000yz^{10}w^{9}-241560692091465866919264967200yz^{9}w^{10}+255278608364499381804113598528yz^{8}w^{11}-108671484975252777892339430256yz^{7}w^{12}-32703255990683041502285859996yz^{6}w^{13}+69384020365110838335373193040yz^{5}w^{14}-41798143441766720557185129792yz^{4}w^{15}+13596744191181025326238500516yz^{3}w^{16}-2398431410918486292277231303yz^{2}w^{17}+182364406052448571440284534yzw^{18}-379876547971164449412082yw^{19}+214256615181640131642573312z^{20}-2349300551129017783471431936z^{19}w+1527325851404972133403052544z^{18}w^{2}+17069342507053251850707787008z^{17}w^{3}-22033705347775598648896014336z^{16}w^{4}-16466775555597829446547619712z^{15}w^{5}+44565178567898695012491088512z^{14}w^{6}-67121338781723577465555884160z^{13}w^{7}+43093664909452873043896904928z^{12}w^{8}+76636240855464416372846478288z^{11}w^{9}-148332573092774998754139034368z^{10}w^{10}+77940647266258181849152880016z^{9}w^{11}+34384033354465784314047056904z^{8}w^{12}-74842461445028458234646304612z^{7}w^{13}+41211305576418642968202199812z^{6}w^{14}-621648087813461035900439460z^{5}w^{15}-11028547846677831320808329454z^{4}w^{16}+6293551129942005525345183815z^{3}w^{17}-1677691909501705353429647942z^{2}w^{18}+217672668547792242594119873zw^{19}-10166891730195206443640125w^{20}}{155985836627141978347008y^{2}z^{18}-1120698072707083591180032y^{2}z^{17}w+3666951967295305957616640y^{2}z^{16}w^{2}-7392772316208639146526720y^{2}z^{15}w^{3}+9509921026882584912495360y^{2}z^{14}w^{4}-5383424009152241080189056y^{2}z^{13}w^{5}-7739247621089832338897280y^{2}z^{12}w^{6}+26368521998811802914168576y^{2}z^{11}w^{7}-41100659927901873061146720y^{2}z^{10}w^{8}+43611605247961312972573104y^{2}z^{9}w^{9}-33714380111752915097253984y^{2}z^{8}w^{10}+18733117842577014683519520y^{2}z^{7}w^{11}-6679412319275734782075768y^{2}z^{6}w^{12}+671366972050517086761924y^{2}z^{5}w^{13}+858179836241512382372724y^{2}z^{4}w^{14}-625303087751676561120576y^{2}z^{3}w^{15}+220756876237231717891374y^{2}z^{2}w^{16}-42352893699427468875675y^{2}zw^{17}+3470502655211233083392y^{2}w^{18}-29263812503124072515328yz^{19}-75027745296366134450688yz^{18}w+224888309305429866776064yz^{17}w^{2}+1666169573473375814822400yz^{16}w^{3}-10370600988707283637905792yz^{15}w^{4}+29713109075589525809513472yz^{14}w^{5}-56393962234103143944440640yz^{13}w^{6}+78039829412550093657489984yz^{12}w^{7}-80554290980982289739652240yz^{11}w^{8}+60231163461267509596048608yz^{10}w^{9}-28063211729189059073506896yz^{9}w^{10}+801448843052821820813424yz^{8}w^{11}+12010875721266134433792780yz^{7}w^{12}-12288037860917504662287888yz^{6}w^{13}+7458195178184447384284374yz^{5}w^{14}-3147558893711003343614166yz^{4}w^{15}+948608837378981692559457yz^{3}w^{16}-200792663261203150390654yz^{2}w^{17}+27769474647233570755358yzw^{18}-1941708403912688140288yw^{19}-27086363792946021765888z^{20}+173542965379150223075328z^{19}w-660239799425451166825728z^{18}w^{2}+2360969362440184246500096z^{17}w^{3}-6989624341486787620600704z^{16}w^{4}+15285626348567967923484672z^{15}w^{5}-24158703210687597991612608z^{14}w^{6}+27158534496131327016446592z^{13}w^{7}-19879880132885347924014672z^{12}w^{8}+5172773340230761558508352z^{11}w^{9}+8217391726341317207572704z^{10}w^{10}-13223243299898334180380400z^{9}w^{11}+10006920397034305589581884z^{8}w^{12}-3935058102656421033056976z^{7}w^{13}-223681949028598639646718z^{6}w^{14}+1414299344604179560313868z^{5}w^{15}-1032515407417392404146407z^{4}w^{16}+448780439706086079019382z^{3}w^{17}-128786773995571652523185z^{2}w^{18}+22668811178819272105835zw^{19}-1838479865759152340992w^{20}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 38.120.4.a.1 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle z$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ -1026X^{6}-684X^{4}Y^{2}+399X^{4}YZ-152X^{4}Z^{2}-189X^{2}Y^{4}+185X^{2}Y^{3}Z-268X^{2}Y^{2}Z^{2}+262X^{2}YZ^{3}-84X^{2}Z^{4}-21Y^{5}Z+8Y^{4}Z^{2}+55Y^{3}Z^{3}-55Y^{2}Z^{4}+14YZ^{5} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{ns}}(2)$ $2$ $120$ $60$ $0$ $0$ full Jacobian
19.120.1-19.a.1.2 $19$ $2$ $2$ $1$ $0$ $1\cdot2$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
19.120.1-19.a.1.2 $19$ $2$ $2$ $1$ $0$ $1\cdot2$
38.80.2-38.a.1.2 $38$ $3$ $3$ $2$ $0$ $2$
38.120.1-19.a.1.2 $38$ $2$ $2$ $1$ $0$ $1\cdot2$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
38.720.16-38.a.1.1 $38$ $3$ $3$ $16$ $0$ $6^{2}$
38.720.16-38.c.1.1 $38$ $3$ $3$ $16$ $0$ $6^{2}$
38.720.16-38.d.1.1 $38$ $3$ $3$ $16$ $0$ $6^{2}$
38.720.22-38.a.2.1 $38$ $3$ $3$ $22$ $0$ $1^{6}\cdot2^{2}\cdot4^{2}$
38.720.22-38.c.1.1 $38$ $3$ $3$ $22$ $4$ $1^{2}\cdot2^{6}\cdot4$
38.720.22-38.e.1.2 $38$ $3$ $3$ $22$ $0$ $6\cdot12$
38.720.22-38.f.1.1 $38$ $3$ $3$ $22$ $0$ $6\cdot12$
38.720.22-38.h.1.1 $38$ $3$ $3$ $22$ $0$ $6\cdot12$
38.4560.157-38.a.1.1 $38$ $19$ $19$ $157$ $20$ $1^{7}\cdot2^{14}\cdot3^{2}\cdot4^{8}\cdot6^{4}\cdot8^{2}\cdot12^{2}\cdot16$
76.480.10-76.a.1.3 $76$ $2$ $2$ $10$ $?$ not computed
76.480.10-76.c.1.1 $76$ $2$ $2$ $10$ $?$ not computed
152.480.10-152.a.1.1 $152$ $2$ $2$ $10$ $?$ not computed
152.480.10-152.c.2.1 $152$ $2$ $2$ $10$ $?$ not computed
228.480.10-228.a.1.1 $228$ $2$ $2$ $10$ $?$ not computed
228.480.10-228.c.2.2 $228$ $2$ $2$ $10$ $?$ not computed