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Properties

Label 12.48.1.n.1
Level $12$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $12$ $\SL_2$-level: $12$ Newform level: $24$
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: 12.48.1.38

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}5&3\\6&5\end{bmatrix}$, $\begin{bmatrix}7&9\\2&5\end{bmatrix}$, $\begin{bmatrix}11&0\\4&11\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^2.D_{12}$
Contains $-I$: yes
Quadratic refinements: 12.96.1-12.n.1.1, 12.96.1-12.n.1.2, 12.96.1-12.n.1.3, 12.96.1-12.n.1.4, 24.96.1-12.n.1.1, 24.96.1-12.n.1.2, 24.96.1-12.n.1.3, 24.96.1-12.n.1.4, 24.96.1-12.n.1.5, 24.96.1-12.n.1.6, 24.96.1-12.n.1.7, 24.96.1-12.n.1.8, 60.96.1-12.n.1.1, 60.96.1-12.n.1.2, 60.96.1-12.n.1.3, 60.96.1-12.n.1.4, 84.96.1-12.n.1.1, 84.96.1-12.n.1.2, 84.96.1-12.n.1.3, 84.96.1-12.n.1.4, 120.96.1-12.n.1.1, 120.96.1-12.n.1.2, 120.96.1-12.n.1.3, 120.96.1-12.n.1.4, 120.96.1-12.n.1.5, 120.96.1-12.n.1.6, 120.96.1-12.n.1.7, 120.96.1-12.n.1.8, 132.96.1-12.n.1.1, 132.96.1-12.n.1.2, 132.96.1-12.n.1.3, 132.96.1-12.n.1.4, 156.96.1-12.n.1.1, 156.96.1-12.n.1.2, 156.96.1-12.n.1.3, 156.96.1-12.n.1.4, 168.96.1-12.n.1.1, 168.96.1-12.n.1.2, 168.96.1-12.n.1.3, 168.96.1-12.n.1.4, 168.96.1-12.n.1.5, 168.96.1-12.n.1.6, 168.96.1-12.n.1.7, 168.96.1-12.n.1.8, 204.96.1-12.n.1.1, 204.96.1-12.n.1.2, 204.96.1-12.n.1.3, 204.96.1-12.n.1.4, 228.96.1-12.n.1.1, 228.96.1-12.n.1.2, 228.96.1-12.n.1.3, 228.96.1-12.n.1.4, 264.96.1-12.n.1.1, 264.96.1-12.n.1.2, 264.96.1-12.n.1.3, 264.96.1-12.n.1.4, 264.96.1-12.n.1.5, 264.96.1-12.n.1.6, 264.96.1-12.n.1.7, 264.96.1-12.n.1.8, 276.96.1-12.n.1.1, 276.96.1-12.n.1.2, 276.96.1-12.n.1.3, 276.96.1-12.n.1.4, 312.96.1-12.n.1.1, 312.96.1-12.n.1.2, 312.96.1-12.n.1.3, 312.96.1-12.n.1.4, 312.96.1-12.n.1.5, 312.96.1-12.n.1.6, 312.96.1-12.n.1.7, 312.96.1-12.n.1.8
Cyclic 12-isogeny field degree: $2$
Cyclic 12-torsion field degree: $8$
Full 12-torsion field degree: $96$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

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

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

$ 0 $ $=$ $ 3 x^{4} - x^{2} y^{2} - 10 x^{2} z^{2} + 3 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 3^3\,\frac{y^{12}+24y^{10}w^{2}+168y^{8}w^{4}+416y^{6}w^{6}+168y^{4}w^{8}+1536y^{2}w^{10}-728z^{12}+8736z^{10}w^{2}-42144z^{8}w^{4}+105056z^{6}w^{6}-143928z^{4}w^{8}+101952z^{2}w^{10}-21392w^{12}}{w^{4}(z^{2}-3w^{2})^{3}(3z^{2}-w^{2})}$

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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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.12.0.k.1 $12$ $4$ $4$ $0$ $0$ full Jacobian
12.24.0.h.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.24.0.i.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.24.1.k.1 $12$ $2$ $2$ $1$ $0$ 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
12.144.5.y.1 $12$ $3$ $3$ $5$ $0$ $1^{4}$
24.96.5.bq.1 $24$ $2$ $2$ $5$ $0$ $1^{4}$
24.96.5.bs.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.96.5.fs.1 $24$ $2$ $2$ $5$ $0$ $1^{4}$
24.96.5.fu.1 $24$ $2$ $2$ $5$ $3$ $1^{4}$
36.144.5.n.1 $36$ $3$ $3$ $5$ $0$ $1^{4}$
36.144.9.cf.1 $36$ $3$ $3$ $9$ $3$ $1^{8}$
36.144.9.cg.1 $36$ $3$ $3$ $9$ $2$ $1^{8}$
60.240.17.kx.1 $60$ $5$ $5$ $17$ $9$ $1^{16}$
60.288.17.gj.1 $60$ $6$ $6$ $17$ $2$ $1^{16}$
60.480.33.ln.1 $60$ $10$ $10$ $33$ $18$ $1^{32}$
120.96.5.gu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.gv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.ma.1 $120$ $2$ $2$ $5$ $?$ not computed
120.96.5.mb.1 $120$ $2$ $2$ $5$ $?$ not computed
168.96.5.ie.1 $168$ $2$ $2$ $5$ $?$ not computed
168.96.5.if.1 $168$ $2$ $2$ $5$ $?$ not computed
168.96.5.na.1 $168$ $2$ $2$ $5$ $?$ not computed
168.96.5.nb.1 $168$ $2$ $2$ $5$ $?$ not computed
264.96.5.fc.1 $264$ $2$ $2$ $5$ $?$ not computed
264.96.5.fd.1 $264$ $2$ $2$ $5$ $?$ not computed
264.96.5.im.1 $264$ $2$ $2$ $5$ $?$ not computed
264.96.5.in.1 $264$ $2$ $2$ $5$ $?$ not computed
312.96.5.fk.1 $312$ $2$ $2$ $5$ $?$ not computed
312.96.5.fl.1 $312$ $2$ $2$ $5$ $?$ not computed
312.96.5.iu.1 $312$ $2$ $2$ $5$ $?$ not computed
312.96.5.iv.1 $312$ $2$ $2$ $5$ $?$ not computed