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Properties

Label 40.72.1.bl.2
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: 40.72.1.19

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}17&39\\34&37\end{bmatrix}$, $\begin{bmatrix}21&18\\0&19\end{bmatrix}$, $\begin{bmatrix}23&16\\34&35\end{bmatrix}$, $\begin{bmatrix}29&20\\12&7\end{bmatrix}$, $\begin{bmatrix}39&2\\38&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.144.1-40.bl.2.1, 40.144.1-40.bl.2.2, 40.144.1-40.bl.2.3, 40.144.1-40.bl.2.4, 40.144.1-40.bl.2.5, 40.144.1-40.bl.2.6, 40.144.1-40.bl.2.7, 40.144.1-40.bl.2.8, 40.144.1-40.bl.2.9, 40.144.1-40.bl.2.10, 40.144.1-40.bl.2.11, 40.144.1-40.bl.2.12, 40.144.1-40.bl.2.13, 40.144.1-40.bl.2.14, 40.144.1-40.bl.2.15, 40.144.1-40.bl.2.16, 120.144.1-40.bl.2.1, 120.144.1-40.bl.2.2, 120.144.1-40.bl.2.3, 120.144.1-40.bl.2.4, 120.144.1-40.bl.2.5, 120.144.1-40.bl.2.6, 120.144.1-40.bl.2.7, 120.144.1-40.bl.2.8, 120.144.1-40.bl.2.9, 120.144.1-40.bl.2.10, 120.144.1-40.bl.2.11, 120.144.1-40.bl.2.12, 120.144.1-40.bl.2.13, 120.144.1-40.bl.2.14, 120.144.1-40.bl.2.15, 120.144.1-40.bl.2.16, 280.144.1-40.bl.2.1, 280.144.1-40.bl.2.2, 280.144.1-40.bl.2.3, 280.144.1-40.bl.2.4, 280.144.1-40.bl.2.5, 280.144.1-40.bl.2.6, 280.144.1-40.bl.2.7, 280.144.1-40.bl.2.8, 280.144.1-40.bl.2.9, 280.144.1-40.bl.2.10, 280.144.1-40.bl.2.11, 280.144.1-40.bl.2.12, 280.144.1-40.bl.2.13, 280.144.1-40.bl.2.14, 280.144.1-40.bl.2.15, 280.144.1-40.bl.2.16
Cyclic 40-isogeny field degree: $4$
Cyclic 40-torsion field degree: $32$
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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^{10}\cdot5^{10}}\cdot\frac{240x^{2}y^{22}-70960000x^{2}y^{20}z^{2}+1466000000000x^{2}y^{18}z^{4}-9288720000000000x^{2}y^{16}z^{6}+28661600000000000000x^{2}y^{14}z^{8}-54688480000000000000000x^{2}y^{12}z^{10}+73439520000000000000000000x^{2}y^{10}z^{12}-71911200000000000000000000000x^{2}y^{8}z^{14}+51923120000000000000000000000000x^{2}y^{6}z^{16}-27318000000000000000000000000000000x^{2}y^{4}z^{18}+9537360000000000000000000000000000000x^{2}y^{2}z^{20}-1996240000000000000000000000000000000000x^{2}z^{22}-25440xy^{22}z+2374560000xy^{20}z^{3}-32140000000000xy^{18}z^{5}+166348320000000000xy^{16}z^{7}-460881600000000000000xy^{14}z^{9}+826498880000000000000000xy^{12}z^{11}-1060451520000000000000000000xy^{10}z^{13}+1000699200000000000000000000000xy^{8}z^{15}-701066720000000000000000000000000xy^{6}z^{17}+357492000000000000000000000000000000xy^{4}z^{19}-121688160000000000000000000000000000000xy^{2}z^{21}+24315040000000000000000000000000000000000xz^{23}-y^{24}+1606160y^{22}z^{2}-59847040000y^{20}z^{4}+498086000000000y^{18}z^{6}-1830661480000000000y^{16}z^{8}+3970762400000000000000y^{14}z^{10}-5957024320000000000000000y^{12}z^{12}+6525638880000000000000000000y^{10}z^{14}-5316351800000000000000000000000y^{8}z^{16}+3233472080000000000000000000000000y^{6}z^{18}-1395048000000000000000000000000000000y^{4}z^{20}+402616240000000000000000000000000000000y^{2}z^{22}-54979960000000000000000000000000000000000z^{24}}{z^{6}y^{4}(y^{2}-1000z^{2})^{2}(x^{2}y^{8}-148000x^{2}y^{6}z^{2}+1082000000x^{2}y^{4}z^{4}-2084000000000x^{2}y^{2}z^{6}+1165000000000000x^{2}z^{8}-106xy^{8}z+3588000xy^{6}z^{3}-17992000000xy^{4}z^{5}+28604000000000xy^{2}z^{7}-14190000000000000xz^{9}+4809y^{8}z^{2}-60032000y^{6}z^{4}+160738000000y^{4}z^{6}-139056000000000y^{2}z^{8}+32085000000000000z^{10})}$

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
$X_{\pm1}(10)$ $10$ $2$ $2$ $0$ $0$ full Jacobian
40.36.0.d.1 $40$ $2$ $2$ $0$ $0$ 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.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.bq.1 $40$ $2$ $2$ $5$ $3$ $1^{2}\cdot2$
40.144.5.ct.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.cx.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.ig.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.il.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.iu.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.je.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.360.13.bz.1 $40$ $5$ $5$ $13$ $3$ $1^{6}\cdot2^{3}$
120.144.5.chd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.chh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cif.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cij.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egs.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.egv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ehu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ehx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.216.13.uj.2 $120$ $3$ $3$ $13$ $?$ not computed
120.288.13.ien.2 $120$ $4$ $4$ $13$ $?$ not computed
200.360.13.bl.2 $200$ $5$ $5$ $13$ $?$ not computed
280.144.5.bfv.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bfw.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgj.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bgk.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bol.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bom.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.boz.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.bpa.1 $280$ $2$ $2$ $5$ $?$ not computed