Properties

Label 40.120.3-10.a.1.2
Level $40$
Index $120$
Genus $3$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $20$ Newform level: $100$
Index: $120$ $\PSL_2$-index:$60$
Genus: $3 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (none of which are rational) Cusp widths $10^{6}$ Cusp orbits $2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10A3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.120.3.21

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&22\\18&11\end{bmatrix}$, $\begin{bmatrix}11&32\\8&19\end{bmatrix}$, $\begin{bmatrix}23&6\\14&17\end{bmatrix}$, $\begin{bmatrix}23&14\\16&39\end{bmatrix}$, $\begin{bmatrix}33&22\\4&37\end{bmatrix}$, $\begin{bmatrix}37&20\\38&23\end{bmatrix}$
Contains $-I$: no $\quad$ (see 10.60.3.a.1 for the level structure with $-I$)
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
Full 40-torsion field degree: $6144$

Jacobian

Conductor: $2^{4}\cdot5^{6}$
Simple: no
Squarefree: no
Decomposition: $1^{3}$
Newforms: 50.2.a.b$^{2}$, 100.2.a.a

Models

Canonical model in $\mathbb{P}^{ 2 }$

$ 0 $ $=$ $ 2 x^{4} - 3 x^{3} y + 3 x^{3} z - 5 x^{2} y^{2} - 18 x^{2} y z - 5 x^{2} z^{2} - 4 x y^{3} - 17 x y^{2} z + \cdots - 2 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 to other modular curves

$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 5^2\,\frac{942686181x^{3}y^{12}+7521205212x^{3}y^{11}z+21841462746x^{3}y^{10}z^{2}+26279261420x^{3}y^{9}z^{3}+8553933195x^{3}y^{8}z^{4}-3838235848x^{3}y^{7}z^{5}-2749142676x^{3}y^{6}z^{6}-3838235848x^{3}y^{5}z^{7}+8553933195x^{3}y^{4}z^{8}+26279261420x^{3}y^{3}z^{9}+21841462746x^{3}y^{2}z^{10}+7521205212x^{3}yz^{11}+942686181x^{3}z^{12}+1161438453x^{2}y^{13}+12366492993x^{2}y^{12}z+50057826552x^{2}y^{11}z^{2}+93833940112x^{2}y^{10}z^{3}+74146673325x^{2}y^{9}z^{4}+6215553441x^{2}y^{8}z^{5}-10741951784x^{2}y^{7}z^{6}+10741951784x^{2}y^{6}z^{7}-6215553441x^{2}y^{5}z^{8}-74146673325x^{2}y^{4}z^{9}-93833940112x^{2}y^{3}z^{10}-50057826552x^{2}y^{2}z^{11}-12366492993x^{2}yz^{12}-1161438453x^{2}z^{13}+816387768xy^{14}+8576491653xy^{13}z+32274478260xy^{12}z^{2}+46701809818xy^{11}z^{3}-1555516012xy^{10}z^{4}-60480360469xy^{9}z^{5}-28296834304xy^{8}z^{6}+12444669420xy^{7}z^{7}-28296834304xy^{6}z^{8}-60480360469xy^{5}z^{9}-1555516012xy^{4}z^{10}+46701809818xy^{3}z^{11}+32274478260xy^{2}z^{12}+8576491653xyz^{13}+816387768xz^{14}+345044610y^{15}+1836929070y^{14}z+2535309000y^{13}z^{2}+365319900y^{12}z^{3}+2866112280y^{11}z^{4}+6122843860y^{10}z^{5}-713164710y^{9}z^{6}-1091471650y^{8}z^{7}+1091471650y^{7}z^{8}+713164710y^{6}z^{9}-6122843860y^{5}z^{10}-2866112280y^{4}z^{11}-365319900y^{3}z^{12}-2535309000y^{2}z^{13}-1836929070yz^{14}-345044610z^{15}}{436429x^{3}y^{12}-32512x^{3}y^{11}z-2136286x^{3}y^{10}z^{2}+3087680x^{3}y^{9}z^{3}-598845x^{3}y^{8}z^{4}-3040832x^{3}y^{7}z^{5}+4626076x^{3}y^{6}z^{6}-3040832x^{3}y^{5}z^{7}-598845x^{3}y^{4}z^{8}+3087680x^{3}y^{3}z^{9}-2136286x^{3}y^{2}z^{10}-32512x^{3}yz^{11}+436429x^{3}z^{12}+537693x^{2}y^{13}+1395853x^{2}y^{12}z-3482508x^{2}y^{11}z^{2}-2075228x^{2}y^{10}z^{3}+11077825x^{2}y^{9}z^{4}-13186479x^{2}y^{8}z^{5}+10001136x^{2}y^{7}z^{6}-10001136x^{2}y^{6}z^{7}+13186479x^{2}y^{5}z^{8}-11077825x^{2}y^{4}z^{9}+2075228x^{2}y^{3}z^{10}+3482508x^{2}y^{2}z^{11}-1395853x^{2}yz^{12}-537693x^{2}z^{13}+377976xy^{14}+927181xy^{13}z-3365160xy^{12}z^{2}-1424414xy^{11}z^{3}+12384616xy^{10}z^{4}-15132733xy^{9}z^{5}+6603592xy^{8}z^{6}-738020xy^{7}z^{7}+6603592xy^{6}z^{8}-15132733xy^{5}z^{9}+12384616xy^{4}z^{10}-1424414xy^{3}z^{11}-3365160xy^{2}z^{12}+927181xyz^{13}+377976xz^{14}+159762y^{15}-436218y^{14}z+105520y^{13}z^{2}+1873220y^{12}z^{3}-5294352y^{11}z^{4}+8188412y^{10}z^{5}-8953166y^{9}z^{6}+8392350y^{8}z^{7}-8392350y^{7}z^{8}+8953166y^{6}z^{9}-8188412y^{5}z^{10}+5294352y^{4}z^{11}-1873220y^{3}z^{12}-105520y^{2}z^{13}+436218yz^{14}-159762z^{15}}$

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
40.12.0-2.a.1.1 $40$ $10$ $10$ $0$ $0$ full Jacobian

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.240.5-10.a.1.2 $40$ $2$ $2$ $5$ $0$ $1^{2}$
40.240.5-10.b.1.2 $40$ $2$ $2$ $5$ $0$ $1^{2}$
40.360.7-10.b.1.1 $40$ $3$ $3$ $7$ $0$ $1^{4}$
40.240.5-20.a.1.1 $40$ $2$ $2$ $5$ $2$ $1^{2}$
40.240.5-20.b.1.2 $40$ $2$ $2$ $5$ $0$ $1^{2}$
40.240.7-20.a.1.3 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.240.7-20.a.1.6 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.240.7-20.b.1.4 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-20.b.1.22 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-20.b.1.23 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-20.c.1.4 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.c.1.5 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.c.1.6 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.d.1.4 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-20.d.1.5 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-20.d.1.6 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-20.e.1.2 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.e.1.6 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.e.1.8 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.f.1.2 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.f.1.6 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.f.1.8 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.g.1.2 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.g.1.6 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.g.1.7 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.h.1.1 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.h.1.6 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-20.h.1.8 $40$ $2$ $2$ $7$ $1$ $1^{4}$
120.240.5-30.a.1.1 $120$ $2$ $2$ $5$ $?$ not computed
120.240.5-30.b.1.4 $120$ $2$ $2$ $5$ $?$ not computed
120.360.13-30.e.1.2 $120$ $3$ $3$ $13$ $?$ not computed
120.480.15-30.a.1.8 $120$ $4$ $4$ $15$ $?$ not computed
40.240.5-40.a.1.1 $40$ $2$ $2$ $5$ $2$ $1^{2}$
40.240.5-40.b.1.1 $40$ $2$ $2$ $5$ $0$ $1^{2}$
40.240.5-40.c.1.3 $40$ $2$ $2$ $5$ $0$ $1^{2}$
40.240.5-40.d.1.3 $40$ $2$ $2$ $5$ $2$ $1^{2}$
40.240.7-40.a.1.3 $40$ $2$ $2$ $7$ $4$ $1^{4}$
40.240.7-40.a.1.8 $40$ $2$ $2$ $7$ $4$ $1^{4}$
40.240.7-40.a.1.10 $40$ $2$ $2$ $7$ $4$ $1^{4}$
40.240.7-40.b.1.3 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-40.b.1.8 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-40.b.1.10 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-40.c.1.3 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-40.c.1.7 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-40.c.1.13 $40$ $2$ $2$ $7$ $1$ $1^{4}$
40.240.7-40.d.1.3 $40$ $2$ $2$ $7$ $3$ $1^{4}$
40.240.7-40.d.1.7 $40$ $2$ $2$ $7$ $3$ $1^{4}$
40.240.7-40.d.1.13 $40$ $2$ $2$ $7$ $3$ $1^{4}$
40.240.7-40.e.1.5 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.240.7-40.e.1.6 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.240.7-40.e.1.12 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.240.7-40.f.1.5 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.240.7-40.f.1.6 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.240.7-40.f.1.12 $40$ $2$ $2$ $7$ $2$ $1^{4}$
40.240.7-40.g.1.2 $40$ $2$ $2$ $7$ $4$ $1^{4}$
40.240.7-40.g.1.3 $40$ $2$ $2$ $7$ $4$ $1^{4}$
40.240.7-40.g.1.12 $40$ $2$ $2$ $7$ $4$ $1^{4}$
40.240.7-40.h.1.2 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-40.h.1.3 $40$ $2$ $2$ $7$ $0$ $1^{4}$
40.240.7-40.h.1.12 $40$ $2$ $2$ $7$ $0$ $1^{4}$
120.240.5-60.a.1.4 $120$ $2$ $2$ $5$ $?$ not computed
120.240.5-60.b.1.3 $120$ $2$ $2$ $5$ $?$ not computed
120.240.7-60.a.1.1 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.a.1.7 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.a.1.16 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.b.1.1 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.b.1.10 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.b.1.15 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.c.1.3 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.c.1.6 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.c.1.15 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.d.1.3 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.d.1.7 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.d.1.14 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.e.1.2 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.e.1.9 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.e.1.16 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.f.1.3 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.f.1.4 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.f.1.9 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.g.1.2 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.g.1.5 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.g.1.13 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.h.1.2 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.h.1.11 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-60.h.1.16 $120$ $2$ $2$ $7$ $?$ not computed
280.240.5-70.a.1.4 $280$ $2$ $2$ $5$ $?$ not computed
280.240.5-70.b.1.4 $280$ $2$ $2$ $5$ $?$ not computed
120.240.5-120.a.1.4 $120$ $2$ $2$ $5$ $?$ not computed
120.240.5-120.b.1.7 $120$ $2$ $2$ $5$ $?$ not computed
120.240.5-120.c.1.3 $120$ $2$ $2$ $5$ $?$ not computed
120.240.5-120.d.1.3 $120$ $2$ $2$ $5$ $?$ not computed
120.240.7-120.a.1.10 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.a.1.11 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.a.1.18 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.b.1.6 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.b.1.23 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.b.1.30 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.c.1.6 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.c.1.21 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.c.1.32 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.d.1.3 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.d.1.6 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.d.1.22 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.e.1.3 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.e.1.18 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.e.1.27 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.f.1.14 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.f.1.27 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.f.1.29 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.g.1.6 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.g.1.29 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.g.1.31 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.h.1.9 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.h.1.14 $120$ $2$ $2$ $7$ $?$ not computed
120.240.7-120.h.1.28 $120$ $2$ $2$ $7$ $?$ not computed
280.240.5-140.a.1.1 $280$ $2$ $2$ $5$ $?$ not computed
280.240.5-140.b.1.4 $280$ $2$ $2$ $5$ $?$ not computed
280.240.7-140.a.1.4 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.a.1.13 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.a.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.b.1.4 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.b.1.11 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.b.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.c.1.7 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.c.1.9 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.c.1.14 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.d.1.5 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.d.1.11 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.d.1.14 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.e.1.1 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.e.1.12 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.e.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.f.1.1 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.f.1.13 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.f.1.14 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.g.1.7 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.g.1.9 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.g.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.h.1.5 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.h.1.11 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-140.h.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.240.5-280.a.1.2 $280$ $2$ $2$ $5$ $?$ not computed
280.240.5-280.b.1.2 $280$ $2$ $2$ $5$ $?$ not computed
280.240.5-280.c.1.5 $280$ $2$ $2$ $5$ $?$ not computed
280.240.5-280.d.1.5 $280$ $2$ $2$ $5$ $?$ not computed
280.240.7-280.a.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.a.1.18 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.a.1.29 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.b.1.15 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.b.1.18 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.b.1.31 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.c.1.3 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.c.1.24 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.c.1.31 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.d.1.3 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.d.1.24 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.d.1.31 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.e.1.10 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.e.1.19 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.e.1.32 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.f.1.10 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.f.1.19 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.f.1.30 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.g.1.6 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.g.1.21 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.g.1.29 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.h.1.6 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.h.1.21 $280$ $2$ $2$ $7$ $?$ not computed
280.240.7-280.h.1.29 $280$ $2$ $2$ $7$ $?$ not computed