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Properties

Label 50.360.4-50.a.2.3
Level $50$
Index $360$
Genus $4$
Analytic rank $0$
Cusps $24$
$\Q$-cusps $4$

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Invariants

Level: $50$ $\SL_2$-level: $50$ Newform level: $50$
Index: $360$ $\PSL_2$-index:$180$
Genus: $4 = 1 + \frac{ 180 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps: $24$ (of which $4$ are rational) Cusp widths $1^{10}\cdot2^{10}\cdot25^{2}\cdot50^{2}$ Cusp orbits $1^{4}\cdot2^{2}\cdot4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3$
$\overline{\Q}$-gonality: $3$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 50F4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 50.360.4.5

Level structure

$\GL_2(\Z/50\Z)$-generators: $\begin{bmatrix}1&18\\0&27\end{bmatrix}$, $\begin{bmatrix}39&37\\0&27\end{bmatrix}$
Contains $-I$: no $\quad$ (see 50.180.4.a.2 for the level structure with $-I$)
Cyclic 50-isogeny field degree: $1$
Cyclic 50-torsion field degree: $10$
Full 50-torsion field degree: $5000$

Jacobian

Conductor: $2^{4}\cdot5^{8}$
Simple: no
Squarefree: yes
Decomposition: $1^{2}\cdot2$
Newforms: 50.2.a.a, 50.2.a.b, 50.2.b.a

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ x z - y w $
$=$ $ - x^{2} w + x y^{2} - y z^{2} - z w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - x^{4} y - x^{3} z^{2} + x y^{3} z - y^{2} z^{3} $
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Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:1:0:0)$, $(0:0:0:1)$, $(1:0:0:0)$, $(0:0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 180 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{x^{30}+18x^{27}yw^{2}+129x^{25}w^{5}+714x^{22}yw^{7}+1902x^{20}w^{10}+5622x^{17}yw^{12}-3642x^{15}w^{15}-88302x^{12}yw^{17}-456399x^{10}w^{20}-2103612x^{7}yw^{22}-5366796x^{5}w^{25}-14448411x^{2}yw^{27}+64y^{30}-1152y^{27}z^{2}w+5952y^{26}zw^{3}-14976y^{25}w^{5}+2688y^{22}z^{2}w^{6}+61056y^{21}zw^{8}+8319y^{20}w^{10}-14700y^{17}z^{2}w^{11}+42354y^{16}zw^{13}+62772y^{15}w^{15}+133191y^{12}z^{2}w^{16}+302880y^{11}zw^{18}+169245y^{10}w^{20}-177123y^{7}z^{2}w^{21}-1446966y^{6}zw^{23}-5006646y^{5}w^{25}-9066123y^{2}z^{2}w^{26}-9080463yzw^{28}+z^{30}-18z^{25}w^{5}+57z^{20}w^{10}+144z^{15}w^{15}+420z^{10}w^{20}+2472z^{5}w^{25}+64w^{30}}{w^{12}(x^{17}y+17x^{15}w^{3}+158x^{12}yw^{5}+805x^{10}w^{8}+4398x^{7}yw^{10}+13905x^{5}w^{13}+38116x^{2}yw^{15}-64y^{12}z^{2}w^{4}+512y^{10}w^{8}+1408y^{7}z^{2}w^{9}+6015y^{6}zw^{11}+14917y^{5}w^{13}+24204y^{2}z^{2}w^{14}+24211yzw^{16}-z^{5}w^{13})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 50.180.4.a.2 :

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ -X^{4}Y+XY^{3}Z-X^{3}Z^{2}-Y^{2}Z^{3} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
10.72.0-10.a.2.4 $10$ $5$ $5$ $0$ $0$ full Jacobian
50.120.0-25.a.1.1 $50$ $3$ $3$ $0$ $0$ 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
50.720.13-50.a.1.2 $50$ $2$ $2$ $13$ $0$ $1^{5}\cdot2^{2}$
50.720.13-50.b.1.1 $50$ $2$ $2$ $13$ $0$ $1^{5}\cdot2^{2}$
50.1800.48-50.g.2.3 $50$ $5$ $5$ $48$ $0$ $4^{3}\cdot8^{4}$
50.1800.48-50.h.1.4 $50$ $5$ $5$ $48$ $0$ $4^{3}\cdot8^{4}$
50.1800.48-50.i.2.4 $50$ $5$ $5$ $48$ $0$ $4^{3}\cdot8^{4}$
50.1800.48-50.j.1.4 $50$ $5$ $5$ $48$ $0$ $4^{3}\cdot8^{4}$
50.1800.48-50.k.1.3 $50$ $5$ $5$ $48$ $12$ $2^{6}\cdot4^{2}\cdot8^{3}$
50.1800.56-50.a.2.3 $50$ $5$ $5$ $56$ $6$ $1^{2}\cdot2^{9}\cdot4^{8}$