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Properties

Label 60.48.1.br.1
Level $60$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

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

Other labels

Cummins and Pauli (CP) label: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.48.1.170

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}2&3\\33&38\end{bmatrix}$, $\begin{bmatrix}35&52\\36&13\end{bmatrix}$, $\begin{bmatrix}37&14\\30&37\end{bmatrix}$, $\begin{bmatrix}38&43\\15&2\end{bmatrix}$, $\begin{bmatrix}46&5\\15&22\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 60.96.1-60.br.1.1, 60.96.1-60.br.1.2, 60.96.1-60.br.1.3, 60.96.1-60.br.1.4, 60.96.1-60.br.1.5, 60.96.1-60.br.1.6, 60.96.1-60.br.1.7, 60.96.1-60.br.1.8, 60.96.1-60.br.1.9, 60.96.1-60.br.1.10, 60.96.1-60.br.1.11, 60.96.1-60.br.1.12, 120.96.1-60.br.1.1, 120.96.1-60.br.1.2, 120.96.1-60.br.1.3, 120.96.1-60.br.1.4, 120.96.1-60.br.1.5, 120.96.1-60.br.1.6, 120.96.1-60.br.1.7, 120.96.1-60.br.1.8, 120.96.1-60.br.1.9, 120.96.1-60.br.1.10, 120.96.1-60.br.1.11, 120.96.1-60.br.1.12, 120.96.1-60.br.1.13, 120.96.1-60.br.1.14, 120.96.1-60.br.1.15, 120.96.1-60.br.1.16, 120.96.1-60.br.1.17, 120.96.1-60.br.1.18, 120.96.1-60.br.1.19, 120.96.1-60.br.1.20, 120.96.1-60.br.1.21, 120.96.1-60.br.1.22, 120.96.1-60.br.1.23, 120.96.1-60.br.1.24, 120.96.1-60.br.1.25, 120.96.1-60.br.1.26, 120.96.1-60.br.1.27, 120.96.1-60.br.1.28
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $192$
Full 60-torsion field degree: $46080$

Jacobian

Conductor: $2^{4}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 48.2.a.a

Models

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

$ 0 $ $=$ $ 10 x y + 5 y^{2} - z^{2} $
$=$ $11 x^{2} - 3 x y + x w - 4 y^{2} + y w + 2 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 25 x^{4} + 5 x^{3} y - x^{2} y^{2} + 60 x^{2} z^{2} + x y z^{2} + 11 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 10w$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{5^3\cdot11^2}\cdot\frac{5240691321354390xz^{10}w-39112973760115750xz^{8}w^{3}+83446566108196500xz^{6}w^{5}-192223714612902500xz^{4}w^{7}+270856187915500000xz^{2}w^{9}-35487355500000xw^{11}+1164244596945705y^{2}z^{10}-4191281054953575y^{2}z^{8}w^{2}-467596157201250y^{2}z^{6}w^{4}+41582992706296250y^{2}z^{4}w^{6}-163302287704187500y^{2}z^{2}w^{8}+198378317778500000y^{2}w^{10}+5240691321354390yz^{10}w-30795723686376550yz^{8}w^{3}+71701880557730500yz^{6}w^{5}-221410649075907500yz^{4}w^{7}+438515933442125000yz^{2}w^{9}-245212366995500000yw^{11}+2570122125034453z^{12}-14396197606084305z^{10}w^{2}+26160504227607750z^{8}w^{4}-12673532903743250z^{6}w^{6}-20552203459136250z^{4}w^{8}+24181473081987500z^{2}w^{10}+9812772843750w^{12}}{z^{2}(84185750xz^{8}w-1622049770xz^{6}w^{3}+695697486xz^{4}w^{5}+1263462750xz^{2}w^{7}+1659771000xw^{9}-1830125y^{2}z^{8}+197620225y^{2}z^{6}w^{2}-1394630875y^{2}z^{4}w^{4}+114914745y^{2}z^{2}w^{6}+120868200y^{2}w^{8}+84185750yz^{8}w-1705503470yz^{6}w^{3}+2241920186yz^{4}w^{5}+832153410yz^{2}w^{7}+1071621900yw^{9}-366025z^{10}+38791995z^{8}w^{2}-218179335z^{6}w^{4}-95319961z^{4}w^{6}-228390210z^{2}w^{8}-483472800w^{10})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(3)$ $3$ $12$ $12$ $0$ $0$ full Jacobian
20.12.0.l.1 $20$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.24.1.l.1 $12$ $2$ $2$ $1$ $0$ dimension zero
20.12.0.l.1 $20$ $4$ $4$ $0$ $0$ full Jacobian
60.24.0.o.1 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.24.0.q.1 $60$ $2$ $2$ $0$ $0$ 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
60.96.3.bw.1 $60$ $2$ $2$ $3$ $0$ $1^{2}$
60.96.3.bx.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.96.3.by.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.96.3.bz.1 $60$ $2$ $2$ $3$ $0$ $1^{2}$
60.144.5.kb.1 $60$ $3$ $3$ $5$ $0$ $1^{4}$
60.240.17.mn.1 $60$ $5$ $5$ $17$ $5$ $1^{16}$
60.288.17.id.1 $60$ $6$ $6$ $17$ $2$ $1^{16}$
60.480.33.oq.1 $60$ $10$ $10$ $33$ $8$ $1^{32}$
120.96.3.vs.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.vt.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.wa.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.wb.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.wy.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.wz.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.xa.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.xb.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.xk.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.xl.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.xo.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.xp.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.5.cc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.ce.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.hs.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.hu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.um.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.uo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.vs.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.vu.1 $120$ $2$ $2$ $5$ $?$ not computed
180.144.5.br.1 $180$ $3$ $3$ $5$ $?$ not computed
180.144.9.dz.1 $180$ $3$ $3$ $9$ $?$ not computed
180.144.9.ev.1 $180$ $3$ $3$ $9$ $?$ not computed