↑

Properties

Label 38.240.4-38.b.1.2
Level $38$
Index $240$
Genus $4$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $38$ $\SL_2$-level: $38$ Newform level: $1444$
Index: $240$ $\PSL_2$-index:$120$
Genus: $4 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 12 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (none of which are rational) Cusp widths $2^{3}\cdot38^{3}$ Cusp orbits $3^{2}$
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: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 38B4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 38.240.4.12

Level structure

$\GL_2(\Z/38\Z)$-generators: $\begin{bmatrix}4&31\\1&37\end{bmatrix}$, $\begin{bmatrix}8&15\\17&33\end{bmatrix}$, $\begin{bmatrix}11&14\\0&31\end{bmatrix}$
Contains $-I$: no $\quad$ (see 38.120.4.b.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: $2^{6}\cdot19^{6}$
Simple: no
Squarefree: no
Decomposition: $1^{4}$
Newforms: 19.2.a.a, 76.2.a.a, 1444.2.a.b$^{2}$

Models

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

$ 0 $ $=$ $ 19 x^{2} - 6 y^{2} - y z + y w + z^{2} - z w + w^{2} $
$=$ $y^{3} - y^{2} z + y^{2} w + y z^{2} - 2 y z w + y w^{2} + z^{2} w - z w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ 361 x^{4} y^{2} - 722 x^{4} y z + 361 x^{4} z^{2} + 722 x^{3} y^{3} - 2166 x^{3} y^{2} z + 2166 x^{3} y z^{2} + \cdots + 20 z^{6} $
 Copy content Toggle raw display

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

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{563365268736y^{2}z^{18}-759682080000y^{2}z^{17}w-24312483526720y^{2}z^{16}w^{2}+129375971369224y^{2}z^{15}w^{3}-254649939445480y^{2}z^{14}w^{4}+80373709601628y^{2}z^{13}w^{5}+657883454074210y^{2}z^{12}w^{6}-1669077918918636y^{2}z^{11}w^{7}+2308139711795782y^{2}z^{10}w^{8}-2269100005846016y^{2}z^{9}w^{9}+1777383131025862y^{2}z^{8}w^{10}-1216974701629356y^{2}z^{7}w^{11}+801237453962770y^{2}z^{6}w^{12}-530568247198692y^{2}z^{5}w^{13}+323916338149160y^{2}z^{4}w^{14}-151992431820776y^{2}z^{3}w^{15}+44030427463040y^{2}z^{2}w^{16}-5419565280000y^{2}zw^{17}-48426219264y^{2}w^{18}-611792372736yz^{19}+7005173552000yz^{18}w-15521589971840yz^{17}w^{2}-70073008988016yz^{16}w^{3}+453159727726461yz^{15}w^{4}-1064507872325470yz^{14}w^{5}+1187757271303578yz^{13}w^{6}-71851716320657yz^{12}w^{7}-2001609410693126yz^{11}w^{8}+3863778780393528yz^{10}w^{9}-4632818501322888yz^{9}w^{10}+4297355972933126yz^{8}w^{11}-3346589845582663yz^{7}w^{12}+2248512599423142yz^{6}w^{13}-1270123653599570yz^{5}w^{14}+551683683295419yz^{4}w^{15}-154208952699504yz^{3}w^{16}+16374178066880yz^{2}w^{17}+2900731196800yzw^{18}-611790603264yw^{19}+884736z^{20}-611808961536z^{19}w+6441969028736z^{18}w^{2}-14151181971584z^{17}w^{3}-51633749676208z^{16}w^{4}+336543266661877z^{15}w^{5}-776553556229958z^{14}w^{6}+883437692252753z^{13}w^{7}-215769522658661z^{12}w^{8}-974880118581634z^{11}w^{9}+1963639989074294z^{10}w^{10}-2296855477046674z^{9}w^{11}+2036958062525299z^{8}w^{12}-1476752599653847z^{7}w^{13}+873921768446202z^{6}w^{14}-392916350028563z^{5}w^{15}+112987307607152z^{4}w^{16}-11565354632384z^{3}w^{17}-2852144232064z^{2}w^{18}+611807192064zw^{19}+884736w^{20}}{316596930y^{2}z^{18}-3368820490y^{2}z^{17}w+17475351197y^{2}z^{16}w^{2}-58058741556y^{2}z^{15}w^{3}+139055947014y^{2}z^{14}w^{4}-258601322194y^{2}z^{13}w^{5}+396192806426y^{2}z^{12}w^{6}-529604255770y^{2}z^{11}w^{7}+650579470271y^{2}z^{10}w^{8}-752593334429y^{2}z^{9}w^{9}+806067901153y^{2}z^{8}w^{10}-765277744426y^{2}z^{7}w^{11}+614284454649y^{2}z^{6}w^{12}-399836655484y^{2}z^{5}w^{13}+202779268026y^{2}z^{4}w^{14}-76351433183y^{2}z^{3}w^{15}+19859575402y^{2}z^{2}w^{16}-3145311134y^{2}zw^{17}+226247370y^{2}w^{18}-90349559yz^{19}+2180799307yz^{18}w-18070075294yz^{17}w^{2}+85899676047yz^{16}w^{3}-281104371598yz^{15}w^{4}+694400940312yz^{14}w^{5}-1369003233785yz^{13}w^{6}+2234604136923yz^{12}w^{7}-3092987022581yz^{11}w^{8}+3677878144540yz^{10}w^{9}-3767589242262yz^{9}w^{10}+3302072259009yz^{8}w^{11}-2437638863768yz^{7}w^{12}+1479673868595yz^{6}w^{13}-713749339019yz^{5}w^{14}+260111083041yz^{4}w^{15}-65730301071yz^{3}w^{16}+9494294105yz^{2}w^{17}-262053405yzw^{18}-90349561yw^{19}-z^{20}-90349583z^{19}w+1864202149z^{18}w^{2}-14610906140z^{17}w^{3}+66876720353z^{16}w^{4}-211893872384z^{15}w^{5}+507265044821z^{14}w^{6}-965716727206z^{13}w^{7}+1509314845081z^{12}w^{8}-1972991707045z^{11}w^{9}+2173780074450z^{10}w^{10}-2014507895221z^{9}w^{11}+1553118945222z^{8}w^{12}-976042556777z^{7}w^{13}+484261488305z^{6}w^{14}-180375297297z^{5}w^{15}+46061167498z^{4}w^{16}-6439331617z^{3}w^{17}+35805807z^{2}w^{18}+90349585zw^{19}-w^{20}}$

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

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

Equation of the image curve:

$0$ $=$ $ 361X^{4}Y^{2}-722X^{4}YZ+361X^{4}Z^{2}+722X^{3}Y^{3}-2166X^{3}Y^{2}Z+2166X^{3}YZ^{2}-722X^{3}Z^{3}+817X^{2}Y^{4}-2394X^{2}Y^{3}Z+3420X^{2}Y^{2}Z^{2}-2242X^{2}YZ^{3}+399X^{2}Z^{4}+456XY^{5}-1406XY^{4}Z+2204XY^{3}Z^{2}-2052XY^{2}Z^{3}+836XYZ^{4}-38XZ^{5}+543Y^{6}-787Y^{5}Z+839Y^{4}Z^{2}-449Y^{3}Z^{3}+274Y^{2}Z^{4}-40YZ^{5}+20Z^{6} $

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.b.1.1 $19$ $2$ $2$ $1$ $0$ $1^{3}$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
19.120.1-19.b.1.1 $19$ $2$ $2$ $1$ $0$ $1^{3}$
38.80.2-38.a.1.2 $38$ $3$ $3$ $2$ $0$ $1^{2}$
38.120.1-19.b.1.1 $38$ $2$ $2$ $1$ $0$ $1^{3}$

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.b.1.2 $38$ $3$ $3$ $16$ $6$ $3^{4}$
38.720.16-38.e.1.2 $38$ $3$ $3$ $16$ $0$ $6^{2}$
38.720.16-38.e.2.1 $38$ $3$ $3$ $16$ $0$ $6^{2}$
38.720.22-38.b.1.2 $38$ $3$ $3$ $22$ $4$ $1^{10}\cdot2^{4}$
38.720.22-38.d.1.1 $38$ $3$ $3$ $22$ $2$ $2^{5}\cdot4^{2}$
38.720.22-38.g.1.2 $38$ $3$ $3$ $22$ $0$ $6\cdot12$
38.720.22-38.g.2.1 $38$ $3$ $3$ $22$ $0$ $6\cdot12$
38.720.22-38.i.1.2 $38$ $3$ $3$ $22$ $6$ $3^{2}\cdot6^{2}$
38.4560.157-38.b.1.1 $38$ $19$ $19$ $157$ $60$ $1^{21}\cdot2^{21}\cdot3^{6}\cdot4^{3}\cdot6^{6}\cdot8^{3}$
76.480.10-76.e.1.4 $76$ $2$ $2$ $10$ $?$ not computed
76.480.10-76.e.2.3 $76$ $2$ $2$ $10$ $?$ not computed
152.480.10-152.e.1.8 $152$ $2$ $2$ $10$ $?$ not computed
152.480.10-152.e.2.6 $152$ $2$ $2$ $10$ $?$ not computed
228.480.10-228.e.1.8 $228$ $2$ $2$ $10$ $?$ not computed
228.480.10-228.e.2.7 $228$ $2$ $2$ $10$ $?$ not computed