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Properties

Label 50.60.2.a.1
Level $50$
Index $60$
Genus $2$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $50$ $\SL_2$-level: $50$ Newform level: $100$
Index: $60$ $\PSL_2$-index:$60$
Genus: $2 = 1 + \frac{ 60 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $2^{5}\cdot50$ Cusp orbits $1^{2}\cdot4$
Elliptic points: $4$ 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: 50A2
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 50.60.2.1

Level structure

$\GL_2(\Z/50\Z)$-generators: $\begin{bmatrix}5&47\\22&45\end{bmatrix}$, $\begin{bmatrix}41&12\\11&45\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 50-isogeny field degree: $3$
Cyclic 50-torsion field degree: $60$
Full 50-torsion field degree: $30000$

Jacobian

Conductor: $2^{4}\cdot5^{4}$
Simple: yes
Squarefree: yes
Decomposition: $2$
Newforms: 100.2.c.a

Models

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

$ 0 $ $=$ $ 3 y^{3} + y^{2} z - y^{2} w + y z^{2} - 2 y w^{2} - z^{2} w - z w^{2} $
$=$ $3 y^{3} + y^{2} z + 2 y^{2} w + y z^{2} + y z w$
$=$ $3 y^{2} z + y z^{2} + 2 y z w + z^{3} + z^{2} w$
$=$ $3 x y^{2} + x y z + 2 x y w + x z^{2} + x z w$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 11 x^{4} + 7 x^{3} z + 10 x^{2} y^{2} + 9 x^{2} z^{2} + 5 x y^{2} z + 3 x z^{3} + z^{4} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{5} + 5x^{3} + 5x - 11 $
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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.

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

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{5}x$
$\displaystyle Z$ $=$ $\displaystyle z$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle -\frac{3}{5}y+\frac{1}{5}z$
$\displaystyle Y$ $=$ $\displaystyle -\frac{2}{25}xy^{2}-\frac{1}{25}xyz$
$\displaystyle Z$ $=$ $\displaystyle \frac{2}{5}y+\frac{1}{5}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{22143375x^{12}+2075941406250x^{2}w^{10}+127163936632yz^{11}-120421393112yz^{10}w+1584514900360yz^{9}w^{2}-3804168482580yz^{8}w^{3}+1302654395760yz^{7}w^{4}-12100540518534yz^{6}w^{5}-42824867140896yz^{5}w^{6}+45770156931090yz^{4}w^{7}-180657350010420yz^{3}w^{8}+172617351532790yz^{2}w^{9}-192421276237123yzw^{10}+53654741476658yw^{11}+16503723673z^{12}+179371882740z^{11}w+1150972852982z^{10}w^{2}+754293722370z^{9}w^{3}+8655872148045z^{8}w^{4}-5904216196386z^{7}w^{5}+18296093787285z^{6}w^{6}-16113953962854z^{5}w^{7}-12375011183070z^{4}w^{8}+31668057026030z^{3}w^{9}-56181374271687z^{2}w^{10}+52776575074975zw^{11}+177147w^{12}}{613924yz^{11}+5859880yz^{10}w+25229950yz^{9}w^{2}+70134990yz^{8}w^{3}+146833485yz^{7}w^{4}+224081997yz^{6}w^{5}+196279545yz^{5}w^{6}-691125yz^{4}w^{7}-213174915yz^{3}w^{8}-235756060yz^{2}w^{9}-113636671yzw^{10}-21484375yw^{11}+89461z^{12}+234126z^{11}w-1332865z^{10}w^{2}-8964525z^{9}w^{3}-30376710z^{8}w^{4}-85006437z^{7}w^{5}-189598812z^{6}w^{6}-296317260z^{5}w^{7}-302957625z^{4}w^{8}-192364165z^{3}w^{9}-68895324z^{2}w^{10}-10653614zw^{11}}$

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
10.12.0.a.2 $10$ $5$ $5$ $0$ $0$ full Jacobian
$X_0(25)$ $25$ $2$ $2$ $0$ $0$ 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
50.120.5.a.2 $50$ $2$ $2$ $5$ $0$ $1^{3}$
50.120.5.b.1 $50$ $2$ $2$ $5$ $0$ $1^{3}$
50.180.6.a.2 $50$ $3$ $3$ $6$ $0$ $1^{2}\cdot2$
50.300.14.b.2 $50$ $5$ $5$ $14$ $0$ $4\cdot8$
50.300.14.d.1 $50$ $5$ $5$ $14$ $0$ $4\cdot8$
50.300.14.f.2 $50$ $5$ $5$ $14$ $0$ $4\cdot8$
50.300.14.h.1 $50$ $5$ $5$ $14$ $0$ $4\cdot8$
50.300.14.j.1 $50$ $5$ $5$ $14$ $4$ $2^{2}\cdot8$
50.300.20.a.1 $50$ $5$ $5$ $20$ $2$ $2^{3}\cdot4^{3}$
100.120.5.c.2 $100$ $2$ $2$ $5$ $?$ not computed
100.120.5.f.2 $100$ $2$ $2$ $5$ $?$ not computed
100.240.13.b.1 $100$ $4$ $4$ $13$ $?$ not computed
150.120.5.c.2 $150$ $2$ $2$ $5$ $?$ not computed
150.120.5.f.2 $150$ $2$ $2$ $5$ $?$ not computed
150.180.10.b.1 $150$ $3$ $3$ $10$ $?$ not computed
150.240.15.f.2 $150$ $4$ $4$ $15$ $?$ not computed
200.120.5.c.2 $200$ $2$ $2$ $5$ $?$ not computed
200.120.5.l.2 $200$ $2$ $2$ $5$ $?$ not computed
200.120.5.o.2 $200$ $2$ $2$ $5$ $?$ not computed
200.120.5.x.2 $200$ $2$ $2$ $5$ $?$ not computed
250.300.20.a.2 $250$ $5$ $5$ $20$ $?$ not computed
300.120.5.i.2 $300$ $2$ $2$ $5$ $?$ not computed
300.120.5.r.2 $300$ $2$ $2$ $5$ $?$ not computed