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Properties

Label 60.48.1.d.1
Level $60$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $4$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $1800$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.48.1.13

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}1&48\\44&29\end{bmatrix}$, $\begin{bmatrix}11&38\\0&31\end{bmatrix}$, $\begin{bmatrix}15&16\\28&3\end{bmatrix}$, $\begin{bmatrix}25&12\\14&47\end{bmatrix}$, $\begin{bmatrix}41&12\\40&43\end{bmatrix}$, $\begin{bmatrix}59&2\\46&15\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.96.1-60.d.1.1, 60.96.1-60.d.1.2, 60.96.1-60.d.1.3, 60.96.1-60.d.1.4, 60.96.1-60.d.1.5, 60.96.1-60.d.1.6, 60.96.1-60.d.1.7, 60.96.1-60.d.1.8, 60.96.1-60.d.1.9, 60.96.1-60.d.1.10, 60.96.1-60.d.1.11, 60.96.1-60.d.1.12, 60.96.1-60.d.1.13, 60.96.1-60.d.1.14, 60.96.1-60.d.1.15, 60.96.1-60.d.1.16, 60.96.1-60.d.1.17, 60.96.1-60.d.1.18, 60.96.1-60.d.1.19, 60.96.1-60.d.1.20, 120.96.1-60.d.1.1, 120.96.1-60.d.1.2, 120.96.1-60.d.1.3, 120.96.1-60.d.1.4, 120.96.1-60.d.1.5, 120.96.1-60.d.1.6, 120.96.1-60.d.1.7, 120.96.1-60.d.1.8, 120.96.1-60.d.1.9, 120.96.1-60.d.1.10, 120.96.1-60.d.1.11, 120.96.1-60.d.1.12, 120.96.1-60.d.1.13, 120.96.1-60.d.1.14, 120.96.1-60.d.1.15, 120.96.1-60.d.1.16, 120.96.1-60.d.1.17, 120.96.1-60.d.1.18, 120.96.1-60.d.1.19, 120.96.1-60.d.1.20
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $192$
Full 60-torsion field degree: $46080$

Jacobian

Conductor: $2^{3}\cdot3^{2}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1800.2.a.m

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 975x - 8750 $
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Rational points

This modular curve has infinitely many rational points, including 4 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^6\cdot5^6}\cdot\frac{240x^{2}y^{14}+119981250x^{2}y^{12}z^{2}+11699083125000x^{2}y^{10}z^{4}+698141978173828125x^{2}y^{8}z^{6}+25165314255322265625000x^{2}y^{6}z^{8}+619858653753626861572265625x^{2}y^{4}z^{10}+9019146589735154342651367187500x^{2}y^{2}z^{12}+76569767265118828289508819580078125x^{2}z^{14}+27300xy^{14}z+6773625000xy^{12}z^{3}+585484674609375xy^{10}z^{5}+31320840931347656250xy^{8}z^{7}+1064641412240332031250000xy^{6}z^{9}+24485121511674499511718750000xy^{4}z^{11}+349143395563227970218658447265625xy^{2}z^{13}+2661114036651611717104911804199218750xz^{15}+y^{16}+1950000y^{14}z^{2}+247784062500y^{12}z^{4}+17154964687500000y^{10}z^{6}+714799505012695312500y^{8}z^{8}+20529202021268554687500000y^{6}z^{10}+375035249606280372619628906250y^{4}z^{12}+4263176617176149677276611328125000y^{2}z^{14}+18954175912072025246679782867431640625z^{16}}{z^{4}y^{4}(180x^{2}y^{6}+46524375x^{2}y^{4}z^{2}+2215704375000x^{2}y^{2}z^{4}+28341790048828125x^{2}z^{6}+15750xy^{6}z+2300400000xy^{4}z^{3}+88070319140625xy^{2}z^{5}+992040499511718750xz^{7}+y^{8}+882000y^{6}z^{2}+62410500000y^{4}z^{4}+1289219414062500y^{2}z^{6}+7087393707275390625z^{8})}$

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
$X_{\pm1}(2,6)$ $6$ $2$ $2$ $0$ $0$ full Jacobian
60.12.0.b.1 $60$ $4$ $4$ $0$ $0$ full Jacobian
60.24.0.t.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.24.1.v.1 $60$ $2$ $2$ $1$ $1$ 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
60.96.1.h.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.96.1.h.2 $60$ $2$ $2$ $1$ $1$ dimension zero
60.96.1.h.3 $60$ $2$ $2$ $1$ $1$ dimension zero
60.96.1.h.4 $60$ $2$ $2$ $1$ $1$ dimension zero
60.96.3.e.1 $60$ $2$ $2$ $3$ $2$ $1^{2}$
60.96.3.f.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.96.3.h.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.96.3.i.1 $60$ $2$ $2$ $3$ $2$ $1^{2}$
60.96.3.r.1 $60$ $2$ $2$ $3$ $1$ $2$
60.96.3.r.2 $60$ $2$ $2$ $3$ $1$ $2$
60.96.3.v.1 $60$ $2$ $2$ $3$ $1$ $2$
60.96.3.v.2 $60$ $2$ $2$ $3$ $1$ $2$
60.144.5.e.1 $60$ $3$ $3$ $5$ $2$ $1^{4}$
60.240.17.h.1 $60$ $5$ $5$ $17$ $3$ $1^{16}$
60.288.17.h.1 $60$ $6$ $6$ $17$ $5$ $1^{16}$
60.480.33.t.1 $60$ $10$ $10$ $33$ $5$ $1^{32}$
120.96.1.lw.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.lw.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.lw.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.lw.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.dy.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.eb.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.eh.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ek.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fu.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fu.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.gn.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.gn.2 $120$ $2$ $2$ $3$ $?$ not computed
180.144.5.d.1 $180$ $3$ $3$ $5$ $?$ not computed
180.144.9.d.1 $180$ $3$ $3$ $9$ $?$ not computed
180.144.9.h.1 $180$ $3$ $3$ $9$ $?$ not computed