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Properties

Label 40.72.1.ba.1
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: 40.72.1.16

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&33\\12&19\end{bmatrix}$, $\begin{bmatrix}5&29\\38&21\end{bmatrix}$, $\begin{bmatrix}11&3\\24&35\end{bmatrix}$, $\begin{bmatrix}27&19\\4&7\end{bmatrix}$, $\begin{bmatrix}31&18\\0&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.144.1-40.ba.1.1, 40.144.1-40.ba.1.2, 40.144.1-40.ba.1.3, 40.144.1-40.ba.1.4, 40.144.1-40.ba.1.5, 40.144.1-40.ba.1.6, 40.144.1-40.ba.1.7, 40.144.1-40.ba.1.8, 40.144.1-40.ba.1.9, 40.144.1-40.ba.1.10, 40.144.1-40.ba.1.11, 40.144.1-40.ba.1.12, 40.144.1-40.ba.1.13, 40.144.1-40.ba.1.14, 40.144.1-40.ba.1.15, 40.144.1-40.ba.1.16, 120.144.1-40.ba.1.1, 120.144.1-40.ba.1.2, 120.144.1-40.ba.1.3, 120.144.1-40.ba.1.4, 120.144.1-40.ba.1.5, 120.144.1-40.ba.1.6, 120.144.1-40.ba.1.7, 120.144.1-40.ba.1.8, 120.144.1-40.ba.1.9, 120.144.1-40.ba.1.10, 120.144.1-40.ba.1.11, 120.144.1-40.ba.1.12, 120.144.1-40.ba.1.13, 120.144.1-40.ba.1.14, 120.144.1-40.ba.1.15, 120.144.1-40.ba.1.16, 280.144.1-40.ba.1.1, 280.144.1-40.ba.1.2, 280.144.1-40.ba.1.3, 280.144.1-40.ba.1.4, 280.144.1-40.ba.1.5, 280.144.1-40.ba.1.6, 280.144.1-40.ba.1.7, 280.144.1-40.ba.1.8, 280.144.1-40.ba.1.9, 280.144.1-40.ba.1.10, 280.144.1-40.ba.1.11, 280.144.1-40.ba.1.12, 280.144.1-40.ba.1.13, 280.144.1-40.ba.1.14, 280.144.1-40.ba.1.15, 280.144.1-40.ba.1.16
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.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
$(0:1:0)$, $(-3:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^5\cdot5^5}\cdot\frac{240x^{2}y^{22}+70960000x^{2}y^{20}z^{2}+530000000000x^{2}y^{18}z^{4}-23773680000000000x^{2}y^{16}z^{6}+38741600000000000000x^{2}y^{14}z^{8}+1048749280000000000000000x^{2}y^{12}z^{10}-1954973280000000000000000000x^{2}y^{10}z^{12}-11646808800000000000000000000000x^{2}y^{8}z^{14}-7973196880000000000000000000000000x^{2}y^{6}z^{16}+1637238000000000000000000000000000000x^{2}y^{4}z^{18}-100262640000000000000000000000000000000x^{2}y^{2}z^{20}+1996240000000000000000000000000000000000x^{2}z^{22}+25440xy^{22}z+2302560000xy^{20}z^{3}-9836000000000xy^{18}z^{5}-453946080000000000xy^{16}z^{7}+2593521600000000000000xy^{14}z^{9}+12594783680000000000000000xy^{12}z^{11}-48443897280000000000000000000xy^{10}z^{13}-165366820800000000000000000000000xy^{8}z^{15}-92500053280000000000000000000000000xy^{6}z^{17}+19663212000000000000000000000000000000xy^{4}z^{19}-1215711840000000000000000000000000000000xy^{2}z^{21}+24315040000000000000000000000000000000000xz^{23}+y^{24}+1606160y^{22}z^{2}+47391040000y^{20}z^{4}-800218000000000y^{18}z^{6}-3059060120000000000y^{16}z^{8}+60027082400000000000000y^{14}z^{10}-10912028480000000000000000y^{12}z^{12}-755874412320000000000000000000y^{10}z^{14}-901749728200000000000000000000000y^{8}z^{16}-103487807920000000000000000000000000y^{6}z^{18}+38036928000000000000000000000000000000y^{4}z^{20}-2621383760000000000000000000000000000000y^{2}z^{22}+54979960000000000000000000000000000000000z^{24}}{z^{3}y^{2}(y^{2}+1000z^{2})^{5}(13000x^{2}y^{6}z+421400000x^{2}y^{4}z^{3}+1703000000000x^{2}y^{2}z^{5}+1525000000000000x^{2}z^{7}+xy^{8}+578000xy^{6}z^{2}+8598400000xy^{4}z^{4}+24958000000000xy^{2}z^{6}+18575000000000000xz^{8}+173y^{8}z+17167000y^{6}z^{3}+111152600000y^{4}z^{5}+153797000000000y^{2}z^{7}+42000000000000000z^{9})}$

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