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Properties

Label 60.48.1.o.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: 60.48.1.75

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}7&16\\29&3\end{bmatrix}$, $\begin{bmatrix}11&8\\19&9\end{bmatrix}$, $\begin{bmatrix}27&28\\56&41\end{bmatrix}$, $\begin{bmatrix}41&56\\7&15\end{bmatrix}$, $\begin{bmatrix}47&44\\39&25\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.96.1-60.o.1.1, 60.96.1-60.o.1.2, 60.96.1-60.o.1.3, 60.96.1-60.o.1.4, 60.96.1-60.o.1.5, 60.96.1-60.o.1.6, 60.96.1-60.o.1.7, 60.96.1-60.o.1.8, 60.96.1-60.o.1.9, 60.96.1-60.o.1.10, 60.96.1-60.o.1.11, 60.96.1-60.o.1.12, 120.96.1-60.o.1.1, 120.96.1-60.o.1.2, 120.96.1-60.o.1.3, 120.96.1-60.o.1.4, 120.96.1-60.o.1.5, 120.96.1-60.o.1.6, 120.96.1-60.o.1.7, 120.96.1-60.o.1.8, 120.96.1-60.o.1.9, 120.96.1-60.o.1.10, 120.96.1-60.o.1.11, 120.96.1-60.o.1.12, 120.96.1-60.o.1.13, 120.96.1-60.o.1.14, 120.96.1-60.o.1.15, 120.96.1-60.o.1.16, 120.96.1-60.o.1.17, 120.96.1-60.o.1.18, 120.96.1-60.o.1.19, 120.96.1-60.o.1.20, 120.96.1-60.o.1.21, 120.96.1-60.o.1.22, 120.96.1-60.o.1.23, 120.96.1-60.o.1.24, 120.96.1-60.o.1.25, 120.96.1-60.o.1.26, 120.96.1-60.o.1.27, 120.96.1-60.o.1.28
Cyclic 60-isogeny field degree: $6$
Cyclic 60-torsion field degree: $96$
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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^3\cdot5^3}\cdot\frac{240x^{2}y^{14}-390318750x^{2}y^{12}z^{2}+192536645625000x^{2}y^{10}z^{4}-14584108278076171875x^{2}y^{8}z^{6}-7081811890072998046875000x^{2}y^{6}z^{8}+47106883895337928619384765625x^{2}y^{4}z^{10}-103952237049392627334594726562500x^{2}y^{2}z^{12}+76569767265118828289508819580078125x^{2}z^{14}+27300xy^{14}z-44438625000xy^{12}z^{3}+22802670924609375xy^{10}z^{5}-3186336440220996093750xy^{8}z^{7}-230220710310859277343750000xy^{6}z^{9}+1609843458434539306640625000000xy^{4}z^{11}-3604874424703494213008880615234375xy^{2}z^{13}+2661114036651611717104911804199218750xz^{15}+y^{16}-480000y^{14}z^{2}-967823437500y^{12}z^{4}+851388165000000000y^{10}z^{6}-193117240477291992187500y^{8}z^{8}-534797705652991699218750000y^{6}z^{10}+9071768050365859840393066406250y^{4}z^{12}-23980154115037041355133056640625000y^{2}z^{14}+18954175912072025246679782867431640625z^{16}}{z^{2}y^{2}(210x^{2}y^{10}+90821250x^{2}y^{8}z^{2}+9194968125000x^{2}y^{6}z^{4}+377824795839843750x^{2}y^{4}z^{6}+6887300202443847656250x^{2}y^{2}z^{8}+46490281332974395751953125x^{2}z^{10}+21075xy^{10}z+5315625000xy^{8}z^{3}+430749298828125xy^{6}z^{5}+15495622433789062500xy^{4}z^{7}+258282340311895751953125xy^{2}z^{9}+1627167827270256042480468750xz^{11}+y^{12}+1418250y^{10}z^{2}+201014156250y^{8}z^{4}+10683315597656250y^{6}z^{6}+261776946557373046875y^{4}z^{8}+2927325036134948730468750y^{2}z^{10}+11622769848647403717041015625z^{12})}$

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
$X_0(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
30.24.0.b.1 $30$ $2$ $2$ $0$ $0$ full Jacobian
60.12.0.g.1 $60$ $4$ $4$ $0$ $0$ 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.n.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.96.1.n.2 $60$ $2$ $2$ $1$ $1$ dimension zero
60.96.1.n.3 $60$ $2$ $2$ $1$ $1$ dimension zero
60.96.1.n.4 $60$ $2$ $2$ $1$ $1$ dimension zero
60.144.5.en.1 $60$ $3$ $3$ $5$ $2$ $1^{4}$
60.240.17.ba.1 $60$ $5$ $5$ $17$ $3$ $1^{16}$
60.288.17.bi.1 $60$ $6$ $6$ $17$ $5$ $1^{16}$
60.480.33.cm.1 $60$ $10$ $10$ $33$ $5$ $1^{32}$
120.96.1.rx.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.rx.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.rx.3 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.rx.4 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.nq.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.nq.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ns.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ns.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pi.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pj.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pu.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pv.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.py.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.pz.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qc.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.qd.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rm.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.rm.2 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ro.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.ro.2 $120$ $2$ $2$ $3$ $?$ not computed
180.144.5.o.1 $180$ $3$ $3$ $5$ $?$ not computed
180.144.9.w.1 $180$ $3$ $3$ $9$ $?$ not computed
180.144.9.be.1 $180$ $3$ $3$ $9$ $?$ not computed