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Properties

Label 40.72.1.ba.2
Level $40$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $2$

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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: $12$ (of which $2$ are rational) Cusp widths $2^{6}\cdot10^{6}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.72.1.23

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&39\\14&33\end{bmatrix}$, $\begin{bmatrix}13&14\\22&5\end{bmatrix}$, $\begin{bmatrix}29&35\\8&11\end{bmatrix}$, $\begin{bmatrix}33&24\\24&23\end{bmatrix}$, $\begin{bmatrix}39&10\\30&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.144.1-40.ba.2.1, 40.144.1-40.ba.2.2, 40.144.1-40.ba.2.3, 40.144.1-40.ba.2.4, 40.144.1-40.ba.2.5, 40.144.1-40.ba.2.6, 40.144.1-40.ba.2.7, 40.144.1-40.ba.2.8, 40.144.1-40.ba.2.9, 40.144.1-40.ba.2.10, 40.144.1-40.ba.2.11, 40.144.1-40.ba.2.12, 40.144.1-40.ba.2.13, 40.144.1-40.ba.2.14, 40.144.1-40.ba.2.15, 40.144.1-40.ba.2.16, 120.144.1-40.ba.2.1, 120.144.1-40.ba.2.2, 120.144.1-40.ba.2.3, 120.144.1-40.ba.2.4, 120.144.1-40.ba.2.5, 120.144.1-40.ba.2.6, 120.144.1-40.ba.2.7, 120.144.1-40.ba.2.8, 120.144.1-40.ba.2.9, 120.144.1-40.ba.2.10, 120.144.1-40.ba.2.11, 120.144.1-40.ba.2.12, 120.144.1-40.ba.2.13, 120.144.1-40.ba.2.14, 120.144.1-40.ba.2.15, 120.144.1-40.ba.2.16, 280.144.1-40.ba.2.1, 280.144.1-40.ba.2.2, 280.144.1-40.ba.2.3, 280.144.1-40.ba.2.4, 280.144.1-40.ba.2.5, 280.144.1-40.ba.2.6, 280.144.1-40.ba.2.7, 280.144.1-40.ba.2.8, 280.144.1-40.ba.2.9, 280.144.1-40.ba.2.10, 280.144.1-40.ba.2.11, 280.144.1-40.ba.2.12, 280.144.1-40.ba.2.13, 280.144.1-40.ba.2.14, 280.144.1-40.ba.2.15, 280.144.1-40.ba.2.16
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.w

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} - 133x - 363 $
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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.

Weierstrass model
$(-3:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2\cdot5}\cdot\frac{6960x^{2}y^{22}-927711760000x^{2}y^{20}z^{2}+16160554000000000x^{2}y^{18}z^{4}-208710943920000000000x^{2}y^{16}z^{6}-17122056960800000000000000x^{2}y^{14}z^{8}-203076472677280000000000000000x^{2}y^{12}z^{10}-1066296947370720000000000000000000x^{2}y^{10}z^{12}-3017495815471200000000000000000000000x^{2}y^{8}z^{14}-4918017723138320000000000000000000000000x^{2}y^{6}z^{16}-4621765135026000000000000000000000000000000x^{2}y^{4}z^{18}-2328491210944560000000000000000000000000000000x^{2}y^{2}z^{20}-487365722656240000000000000000000000000000000000x^{2}z^{22}-16326240xy^{22}z+368900640000xy^{20}z^{3}+478698668000000000xy^{18}z^{5}-14348870927520000000000xy^{16}z^{7}-454057207396800000000000000xy^{14}z^{9}-4170840422079680000000000000000xy^{12}z^{11}-18741283319814720000000000000000000xy^{10}z^{13}-47435996750011200000000000000000000000xy^{8}z^{15}-70987597093053920000000000000000000000000xy^{6}z^{17}-62333862311388000000000000000000000000000000xy^{4}z^{19}-29708862304659360000000000000000000000000000000xy^{2}z^{21}-5936279296875040000000000000000000000000000000000xz^{23}-y^{24}+13296786640y^{22}z^{2}+451808905760000y^{20}z^{4}+4142318878000000000y^{18}z^{6}-553681023102280000000000y^{16}z^{8}-9189407136735200000000000000y^{14}z^{10}-59154453093387520000000000000000y^{12}z^{12}-197963938114507680000000000000000000y^{10}z^{14}-380293538654967800000000000000000000000y^{8}z^{16}-432152765539216880000000000000000000000000y^{6}z^{18}-284169128623284000000000000000000000000000000y^{4}z^{20}-98291015624145040000000000000000000000000000000y^{2}z^{22}-13422546386719960000000000000000000000000000000000z^{24}}{y^{2}(y^{2}+1000z^{2})(x^{2}y^{18}+12665000x^{2}y^{16}z^{2}+2964610000000x^{2}y^{14}z^{4}+53311862000000000x^{2}y^{12}z^{6}+165735924000000000000x^{2}y^{10}z^{8}+126904328000000000000000x^{2}y^{8}z^{10}-6610000000000000000000x^{2}y^{6}z^{12}-630000000000000000000000x^{2}y^{4}z^{14}-37000000000000000000000000x^{2}y^{2}z^{16}-1000000000000000000000000000x^{2}z^{18}+466xy^{18}z+1083040000xy^{16}z^{3}+100872780000000xy^{14}z^{5}+1001251932000000000xy^{12}z^{7}+2364977104000000000000xy^{10}z^{9}+1543237548000000000000000xy^{8}z^{11}+33140000000000000000000xy^{6}z^{13}+2900000000000000000000000xy^{4}z^{15}+158000000000000000000000000xy^{2}z^{17}+4000000000000000000000000000xz^{19}+98889y^{18}z^{2}+65356535000y^{16}z^{4}+2440548750000000y^{14}z^{6}+11408207138000000000y^{12}z^{8}+13437192096000000000000y^{10}z^{10}+3491646192000000000000000y^{8}z^{12}+749010000000000000000000y^{6}z^{14}+73470000000000000000000000y^{4}z^{16}+4407000000000000000000000000y^{2}z^{18}+121000000000000000000000000000z^{20})}$

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.a.1 $10$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.cj.2 $40$ $3$ $3$ $1$ $0$ dimension zero
40.36.0.f.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.1.e.1 $40$ $2$ $2$ $1$ $0$ 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.ek.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.es.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.fm.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.fu.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.hw.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.ic.2 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.5.iy.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.je.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.360.13.cl.1 $40$ $5$ $5$ $13$ $2$ $1^{6}\cdot2^{3}$
120.144.5.clo.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cme.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cns.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.coi.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ecq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.edc.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eeu.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.efg.2 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.su.1 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.ids.1 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.ba.2 $200$ $5$ $5$ $13$ $?$ not computed
280.144.5.bbw.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bca.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bcy.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bdc.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bkm.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bkq.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.blo.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bls.2 $280$ $2$ $2$ $5$ $?$ not computed