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Properties

Label 60.24.1.v.1
Level $60$
Index $24$
Genus $1$
Analytic rank $1$
Cusps $4$
$\Q$-cusps $4$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $1800$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (all of which are rational) Cusp widths $2\cdot4\cdot6\cdot12$ Cusp orbits $1^{4}$
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: 12F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.24.1.9

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}2&29\\15&52\end{bmatrix}$, $\begin{bmatrix}5&56\\12&25\end{bmatrix}$, $\begin{bmatrix}17&26\\12&55\end{bmatrix}$, $\begin{bmatrix}52&3\\9&38\end{bmatrix}$, $\begin{bmatrix}53&30\\18&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.48.1-60.v.1.1, 60.48.1-60.v.1.2, 60.48.1-60.v.1.3, 60.48.1-60.v.1.4, 60.48.1-60.v.1.5, 60.48.1-60.v.1.6, 60.48.1-60.v.1.7, 60.48.1-60.v.1.8, 60.48.1-60.v.1.9, 60.48.1-60.v.1.10, 60.48.1-60.v.1.11, 60.48.1-60.v.1.12, 60.48.1-60.v.1.13, 60.48.1-60.v.1.14, 60.48.1-60.v.1.15, 60.48.1-60.v.1.16, 120.48.1-60.v.1.1, 120.48.1-60.v.1.2, 120.48.1-60.v.1.3, 120.48.1-60.v.1.4, 120.48.1-60.v.1.5, 120.48.1-60.v.1.6, 120.48.1-60.v.1.7, 120.48.1-60.v.1.8, 120.48.1-60.v.1.9, 120.48.1-60.v.1.10, 120.48.1-60.v.1.11, 120.48.1-60.v.1.12, 120.48.1-60.v.1.13, 120.48.1-60.v.1.14, 120.48.1-60.v.1.15, 120.48.1-60.v.1.16
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $192$
Full 60-torsion field degree: $92160$

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} - 5475x + 148750 $
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Rational points

This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^6\cdot5^6}\cdot\frac{60x^{2}y^{6}-15339375x^{2}y^{4}z^{2}+1047026250000x^{2}y^{2}z^{4}-23666308681640625x^{2}z^{6}-7350xy^{6}z+1306125000xy^{4}z^{3}-89389353515625xy^{2}z^{5}+2040759965917968750xz^{7}-y^{8}+312000y^{6}z^{2}-35751375000y^{4}z^{4}+2000370304687500y^{2}z^{6}-42966811672119140625z^{8}}{z^{4}y^{2}(120x^{2}-10200xz-y^{2}+210000z^{2})}$

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
$X_0(6)$ $6$ $2$ $2$ $0$ $0$ full Jacobian
60.6.0.d.1 $60$ $4$ $4$ $0$ $0$ 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
60.48.1.d.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.48.1.f.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.48.1.n.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.48.1.o.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.48.1.r.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.48.1.s.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.48.1.v.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.48.1.w.1 $60$ $2$ $2$ $1$ $1$ dimension zero
60.72.3.ld.1 $60$ $3$ $3$ $3$ $2$ $1^{2}$
60.120.9.dm.1 $60$ $5$ $5$ $9$ $2$ $1^{8}$
60.144.9.fq.1 $60$ $6$ $6$ $9$ $3$ $1^{8}$
60.240.17.na.1 $60$ $10$ $10$ $17$ $4$ $1^{16}$
120.48.1.gj.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.kb.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.zn.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.zq.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bah.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bak.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.bat.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.48.1.baw.1 $120$ $2$ $2$ $1$ $?$ dimension zero
180.72.3.be.1 $180$ $3$ $3$ $3$ $?$ not computed
180.72.5.n.1 $180$ $3$ $3$ $5$ $?$ not computed
180.72.5.r.1 $180$ $3$ $3$ $5$ $?$ not computed