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Properties

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

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Invariants

Level: $40$ $\SL_2$-level: $20$ 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 $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.72.1.28

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&20\\0&31\end{bmatrix}$, $\begin{bmatrix}9&12\\38&13\end{bmatrix}$, $\begin{bmatrix}9&37\\28&3\end{bmatrix}$, $\begin{bmatrix}23&20\\14&9\end{bmatrix}$, $\begin{bmatrix}29&37\\10&11\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.144.1-40.bl.1.1, 40.144.1-40.bl.1.2, 40.144.1-40.bl.1.3, 40.144.1-40.bl.1.4, 40.144.1-40.bl.1.5, 40.144.1-40.bl.1.6, 40.144.1-40.bl.1.7, 40.144.1-40.bl.1.8, 40.144.1-40.bl.1.9, 40.144.1-40.bl.1.10, 40.144.1-40.bl.1.11, 40.144.1-40.bl.1.12, 40.144.1-40.bl.1.13, 40.144.1-40.bl.1.14, 40.144.1-40.bl.1.15, 40.144.1-40.bl.1.16, 120.144.1-40.bl.1.1, 120.144.1-40.bl.1.2, 120.144.1-40.bl.1.3, 120.144.1-40.bl.1.4, 120.144.1-40.bl.1.5, 120.144.1-40.bl.1.6, 120.144.1-40.bl.1.7, 120.144.1-40.bl.1.8, 120.144.1-40.bl.1.9, 120.144.1-40.bl.1.10, 120.144.1-40.bl.1.11, 120.144.1-40.bl.1.12, 120.144.1-40.bl.1.13, 120.144.1-40.bl.1.14, 120.144.1-40.bl.1.15, 120.144.1-40.bl.1.16, 280.144.1-40.bl.1.1, 280.144.1-40.bl.1.2, 280.144.1-40.bl.1.3, 280.144.1-40.bl.1.4, 280.144.1-40.bl.1.5, 280.144.1-40.bl.1.6, 280.144.1-40.bl.1.7, 280.144.1-40.bl.1.8, 280.144.1-40.bl.1.9, 280.144.1-40.bl.1.10, 280.144.1-40.bl.1.11, 280.144.1-40.bl.1.12, 280.144.1-40.bl.1.13, 280.144.1-40.bl.1.14, 280.144.1-40.bl.1.15, 280.144.1-40.bl.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

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - 133x + 363 $
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Rational points

This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.

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}{2^2\cdot5^2}\cdot\frac{240x^{2}y^{22}-3303760000x^{2}y^{20}z^{2}+4122904400000000x^{2}y^{18}z^{4}-611815615920000000000x^{2}y^{16}z^{6}+18703076117600000000000000x^{2}y^{14}z^{8}-205181788475680000000000000000x^{2}y^{12}z^{10}+1067206027362720000000000000000000x^{2}y^{10}z^{12}-3017497556200800000000000000000000000x^{2}y^{8}z^{14}+4918017578022320000000000000000000000000x^{2}y^{6}z^{16}-4621765136710800000000000000000000000000000x^{2}y^{4}z^{18}+2328491210937360000000000000000000000000000000x^{2}y^{2}z^{20}-487365722656240000000000000000000000000000000000x^{2}z^{22}-97440xy^{22}z+415971360000xy^{20}z^{3}-273270570400000000xy^{18}z^{5}+22279226543520000000000xy^{16}z^{7}-478300833441600000000000000xy^{14}z^{9}+4199018264406080000000000000000xy^{12}z^{11}-18752396618886720000000000000000000xy^{10}z^{13}+47435991221620800000000000000000000000xy^{8}z^{15}-70987597656741920000000000000000000000000xy^{6}z^{17}+62333862304720800000000000000000000000000000xy^{4}z^{19}-29708862304688160000000000000000000000000000000xy^{2}z^{21}+5936279296875040000000000000000000000000000000000xz^{23}-y^{24}+16942160y^{22}z^{2}-47017462240000y^{20}z^{4}+14062065311600000000y^{18}z^{6}-653562095742280000000000y^{16}z^{8}+9367476268858400000000000000y^{14}z^{10}-59276474853541120000000000000000y^{12}z^{12}+197989740422515680000000000000000000y^{10}z^{14}-380293318152030200000000000000000000000y^{8}z^{16}+432152783214964880000000000000000000000000y^{6}z^{18}-284169128418919200000000000000000000000000000y^{4}z^{20}+98291015625016240000000000000000000000000000000y^{2}z^{22}-13422546386719960000000000000000000000000000000000z^{24}}{zy^{4}(17600x^{2}y^{16}z+69600000x^{2}y^{14}z^{3}+157900000000x^{2}y^{12}z^{5}-221800000000000x^{2}y^{10}z^{7}+86500000000000000x^{2}y^{8}z^{9}-2800000000000000000x^{2}y^{6}z^{11}-5900000000000000000000x^{2}y^{4}z^{13}+1400000000000000000000000x^{2}y^{2}z^{15}-100000000000000000000000000x^{2}z^{17}+xy^{18}-666600xy^{16}z^{2}-1341600000xy^{14}z^{4}+2522600000000xy^{12}z^{6}-1425200000000000xy^{10}z^{8}+263000000000000000xy^{8}z^{10}+64800000000000000000xy^{6}z^{12}-38600000000000000000000xy^{4}z^{14}+6600000000000000000000000xy^{2}z^{16}-400000000000000000000000000xz^{18}-213y^{18}z+8391400y^{16}z^{3}-26721600000y^{14}z^{5}-5478900000000y^{12}z^{7}+25211800000000000y^{10}z^{9}-11987500000000000000y^{8}z^{11}+1030800000000000000000y^{6}z^{13}+598900000000000000000000y^{4}z^{15}-162400000000000000000000000y^{2}z^{17}+12100000000000000000000000000z^{19})}$

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
40.36.0.d.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.36.1.g.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.m.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.bq.2 $40$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
40.144.5.ct.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.cx.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.ig.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.il.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.iu.2 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.je.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.360.13.bz.1 $40$ $5$ $5$ $13$ $3$ $1^{6}\cdot2^{3}$
120.144.5.chd.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.chh.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cif.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cij.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egs.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egv.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ehu.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ehx.2 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.uj.1 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.ien.1 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.bl.1 $200$ $5$ $5$ $13$ $?$ not computed
280.144.5.bfv.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bfw.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgj.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgk.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bol.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bom.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.boz.2 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bpa.2 $280$ $2$ $2$ $5$ $?$ not computed