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Properties

Label 40.72.1.bj.1
Level $40$
Index $72$
Genus $1$
Analytic rank $1$
Cusps $12$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $10$ Newform level: $1600$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (none of which are rational) Cusp widths $2^{6}\cdot10^{6}$ Cusp orbits $2^{4}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.72.1.34

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}9&0\\38&31\end{bmatrix}$, $\begin{bmatrix}11&16\\10&7\end{bmatrix}$, $\begin{bmatrix}27&6\\8&15\end{bmatrix}$, $\begin{bmatrix}31&27\\24&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.144.1-40.bj.1.1, 80.144.1-40.bj.1.2, 80.144.1-40.bj.1.3, 80.144.1-40.bj.1.4, 80.144.1-40.bj.1.5, 80.144.1-40.bj.1.6, 80.144.1-40.bj.1.7, 80.144.1-40.bj.1.8, 80.144.1-40.bj.1.9, 80.144.1-40.bj.1.10, 80.144.1-40.bj.1.11, 80.144.1-40.bj.1.12, 80.144.1-40.bj.1.13, 80.144.1-40.bj.1.14, 80.144.1-40.bj.1.15, 80.144.1-40.bj.1.16, 240.144.1-40.bj.1.1, 240.144.1-40.bj.1.2, 240.144.1-40.bj.1.3, 240.144.1-40.bj.1.4, 240.144.1-40.bj.1.5, 240.144.1-40.bj.1.6, 240.144.1-40.bj.1.7, 240.144.1-40.bj.1.8, 240.144.1-40.bj.1.9, 240.144.1-40.bj.1.10, 240.144.1-40.bj.1.11, 240.144.1-40.bj.1.12, 240.144.1-40.bj.1.13, 240.144.1-40.bj.1.14, 240.144.1-40.bj.1.15, 240.144.1-40.bj.1.16
Cyclic 40-isogeny field degree: $4$
Cyclic 40-torsion field degree: $64$
Full 40-torsion field degree: $10240$

Jacobian

Conductor: $2^{6}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1600.2.a.c

Models

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

$ 0 $ $=$ $ 5 x^{2} - 4 x y + 5 x z + y^{2} - 2 y z $
$=$ $10 x^{2} - 5 x z - y^{2} + 10 y z - 20 z^{2} + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 4 x^{4} - 16 x^{3} z - 10 x^{2} y^{2} + 28 x^{2} z^{2} + 20 x y^{2} z - 20 x z^{3} - 10 y^{2} z^{2} + 5 z^{4} $
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Rational points

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

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{5}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{2}{5}y$

Maps to other modular curves

$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^6\,\frac{2157353518104850503164062500xz^{17}-8240610389605239618534375000xz^{15}w^{2}+10090050458388924757682812500xz^{13}w^{4}-5371750959362823154563437500xz^{11}w^{6}+1156531282440773126519937500xz^{9}w^{8}+6880947707717847603905000xz^{7}w^{10}-32587171972752993172750875xz^{5}w^{12}+2401238332777150628605350xz^{3}w^{14}-28465922601429820568685xzw^{16}+844514236291057675607812500y^{2}z^{16}-1913413639986707471071875000y^{2}z^{14}w^{2}+1177581242129306950881562500y^{2}z^{12}w^{4}-25516419153997625560687500y^{2}z^{10}w^{6}-190410201146212403667512500y^{2}z^{8}w^{8}+56789001060804098272069000y^{2}z^{6}w^{10}-4958412272466685967657775y^{2}z^{4}w^{12}+134416668944325398054190y^{2}z^{2}w^{14}-488054203184657927961y^{2}w^{16}-3036338932758035152078125000yz^{17}+5753186603978101215918750000yz^{15}w^{2}-2607110515771944460565625000yz^{13}w^{4}-891363786875261322163125000yz^{11}w^{6}+1076604541069783919630125000yz^{9}w^{8}-327962281953233119573810000yz^{7}w^{10}+41235018860908532473953750yz^{5}w^{12}-2059200859737799319462700yz^{3}w^{14}+24652202339377681048890yzw^{16}+844087635878418218531250000z^{18}+3673036475130808186790625000z^{16}w^{2}-9409541668176637492537500000z^{14}w^{4}+7792478443964685605558125000z^{12}w^{6}-3032974729985092329887125000z^{10}w^{8}+585946582446089985790475000z^{8}w^{10}-53368906438162556472913500z^{6}w^{12}+2259734574746707654422450z^{4}w^{14}-46277028923223727363800z^{2}w^{16}+245295497263642977242w^{18}}{79901982152031500117187500xz^{17}-103830739842460411865625000xz^{15}w^{2}+21051568794651809443046875xz^{13}w^{4}+30589487651066684531546875xz^{11}w^{6}-21534177275186842456206250xz^{9}w^{8}+5664108277979407384277500xz^{7}w^{10}-626179297601065785343000xz^{5}w^{12}+15940819203312387659600xz^{3}w^{14}+872512063533028678240xzw^{16}+31278305047816950948437500y^{2}z^{16}-46054954852640510278125000y^{2}z^{14}w^{2}+22787964681048763921359375y^{2}z^{12}w^{4}-2674220493840808266290625y^{2}z^{10}w^{6}-1266269349741179007916250y^{2}z^{8}w^{8}+440501646044154153907500y^{2}z^{6}w^{10}-44819097271038605920600y^{2}z^{4}w^{12}+955966174306614225040y^{2}z^{2}w^{14}+15466266033958700896y^{2}w^{16}-112456997509556857484375000yz^{17}+209576011581838381581250000yz^{15}w^{2}-143284512552340687293593750yz^{13}w^{4}+39561304408564528824406250yz^{11}w^{6}-329400541859556878587500yz^{9}w^{8}-2085263942273429645195000yz^{7}w^{10}+388138116465625748222000yz^{5}w^{12}-19874597585866994890400yz^{3}w^{14}-146627388051810644160yzw^{16}+31262505032534008093750000z^{18}-149241455127749539009375000z^{16}w^{2}+189576894361928959465937500z^{14}w^{4}-109023678133535040338531250z^{12}w^{6}+32068124141243711866718750z^{10}w^{8}-4622631863070688224322500z^{8}w^{10}+212645673195403942767000z^{6}w^{12}+15724225470082795603600z^{4}w^{14}-1522770852233722039520z^{2}w^{16}+15498242965359114688w^{18}}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
10.36.0.b.1 $10$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.ct.2 $40$ $3$ $3$ $1$ $1$ dimension zero
40.36.0.b.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.1.f.1 $40$ $2$ $2$ $1$ $1$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.144.5.ff.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.fj.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.gh.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.gl.1 $40$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
40.144.5.iq.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.it.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.js.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.jv.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.360.13.ct.1 $40$ $5$ $5$ $13$ $3$ $1^{6}\cdot2^{3}$
120.144.5.cnj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cnn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cpn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cpr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eei.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eel.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egm.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egp.1 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.uh.2 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.iel.2 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.bj.1 $200$ $5$ $5$ $13$ $?$ not computed
280.144.5.bfc.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bfd.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bfq.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bfr.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bns.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bnt.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bog.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.boh.1 $280$ $2$ $2$ $5$ $?$ not computed