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Properties

Label 12.48.1.f.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: $72$
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: 12.48.1.42

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}1&5\\6&1\end{bmatrix}$, $\begin{bmatrix}1&10\\0&1\end{bmatrix}$, $\begin{bmatrix}5&11\\6&5\end{bmatrix}$, $\begin{bmatrix}11&9\\0&1\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^2\times D_{12}$
Contains $-I$: yes
Quadratic refinements: 12.96.1-12.f.1.1, 12.96.1-12.f.1.2, 12.96.1-12.f.1.3, 12.96.1-12.f.1.4, 12.96.1-12.f.1.5, 12.96.1-12.f.1.6, 24.96.1-12.f.1.1, 24.96.1-12.f.1.2, 24.96.1-12.f.1.3, 24.96.1-12.f.1.4, 24.96.1-12.f.1.5, 24.96.1-12.f.1.6, 60.96.1-12.f.1.1, 60.96.1-12.f.1.2, 60.96.1-12.f.1.3, 60.96.1-12.f.1.4, 60.96.1-12.f.1.5, 60.96.1-12.f.1.6, 84.96.1-12.f.1.1, 84.96.1-12.f.1.2, 84.96.1-12.f.1.3, 84.96.1-12.f.1.4, 84.96.1-12.f.1.5, 84.96.1-12.f.1.6, 120.96.1-12.f.1.1, 120.96.1-12.f.1.2, 120.96.1-12.f.1.3, 120.96.1-12.f.1.4, 120.96.1-12.f.1.5, 120.96.1-12.f.1.6, 132.96.1-12.f.1.1, 132.96.1-12.f.1.2, 132.96.1-12.f.1.3, 132.96.1-12.f.1.4, 132.96.1-12.f.1.5, 132.96.1-12.f.1.6, 156.96.1-12.f.1.1, 156.96.1-12.f.1.2, 156.96.1-12.f.1.3, 156.96.1-12.f.1.4, 156.96.1-12.f.1.5, 156.96.1-12.f.1.6, 168.96.1-12.f.1.1, 168.96.1-12.f.1.2, 168.96.1-12.f.1.3, 168.96.1-12.f.1.4, 168.96.1-12.f.1.5, 168.96.1-12.f.1.6, 204.96.1-12.f.1.1, 204.96.1-12.f.1.2, 204.96.1-12.f.1.3, 204.96.1-12.f.1.4, 204.96.1-12.f.1.5, 204.96.1-12.f.1.6, 228.96.1-12.f.1.1, 228.96.1-12.f.1.2, 228.96.1-12.f.1.3, 228.96.1-12.f.1.4, 228.96.1-12.f.1.5, 228.96.1-12.f.1.6, 264.96.1-12.f.1.1, 264.96.1-12.f.1.2, 264.96.1-12.f.1.3, 264.96.1-12.f.1.4, 264.96.1-12.f.1.5, 264.96.1-12.f.1.6, 276.96.1-12.f.1.1, 276.96.1-12.f.1.2, 276.96.1-12.f.1.3, 276.96.1-12.f.1.4, 276.96.1-12.f.1.5, 276.96.1-12.f.1.6, 312.96.1-12.f.1.1, 312.96.1-12.f.1.2, 312.96.1-12.f.1.3, 312.96.1-12.f.1.4, 312.96.1-12.f.1.5, 312.96.1-12.f.1.6
Cyclic 12-isogeny field degree: $2$
Cyclic 12-torsion field degree: $8$
Full 12-torsion field degree: $96$

Jacobian

Conductor: $2^{3}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 72.2.a.a

Models

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

$ 0 $ $=$ $ 3 x y - z^{2} $
$=$ $x^{2} - x y + 9 y^{2} - 3 z^{2} + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{4} + 3 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3}\cdot\frac{(4z^{2}-3w^{2})(279552y^{2}z^{8}+39168y^{2}z^{6}w^{2}-98496y^{2}z^{4}w^{4}+589680y^{2}z^{2}w^{6}-176904y^{2}w^{8}-10240z^{10}-9216z^{8}w^{2}+46656z^{6}w^{4}-212112z^{4}w^{6}+131220z^{2}w^{8}-19683w^{10})}{w^{2}z^{4}(24y^{2}z^{4}+18y^{2}z^{2}w^{2}-27y^{2}w^{4}-8z^{6}-3z^{4}w^{2})}$

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
12.12.0.c.1 $12$ $4$ $4$ $0$ $0$ full Jacobian
12.24.0.d.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.24.0.i.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.24.1.i.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
12.96.3.k.1 $12$ $2$ $2$ $3$ $0$ $2$
12.96.3.k.2 $12$ $2$ $2$ $3$ $0$ $2$
12.144.5.l.1 $12$ $3$ $3$ $5$ $0$ $1^{4}$
24.96.3.dp.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.dp.2 $24$ $2$ $2$ $3$ $0$ $2$
36.144.5.f.1 $36$ $3$ $3$ $5$ $0$ $1^{4}$
36.144.9.k.1 $36$ $3$ $3$ $9$ $3$ $1^{8}$
36.144.9.l.1 $36$ $3$ $3$ $9$ $2$ $1^{8}$
60.96.3.x.1 $60$ $2$ $2$ $3$ $0$ $2$
60.96.3.x.2 $60$ $2$ $2$ $3$ $0$ $2$
60.240.17.j.1 $60$ $5$ $5$ $17$ $6$ $1^{16}$
60.288.17.n.1 $60$ $6$ $6$ $17$ $3$ $1^{16}$
60.480.33.bd.1 $60$ $10$ $10$ $33$ $13$ $1^{32}$
84.96.3.x.1 $84$ $2$ $2$ $3$ $?$ not computed
84.96.3.x.2 $84$ $2$ $2$ $3$ $?$ not computed
120.96.3.in.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.in.2 $120$ $2$ $2$ $3$ $?$ not computed
132.96.3.x.1 $132$ $2$ $2$ $3$ $?$ not computed
132.96.3.x.2 $132$ $2$ $2$ $3$ $?$ not computed
156.96.3.x.1 $156$ $2$ $2$ $3$ $?$ not computed
156.96.3.x.2 $156$ $2$ $2$ $3$ $?$ not computed
168.96.3.gr.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.gr.2 $168$ $2$ $2$ $3$ $?$ not computed
204.96.3.x.1 $204$ $2$ $2$ $3$ $?$ not computed
204.96.3.x.2 $204$ $2$ $2$ $3$ $?$ not computed
228.96.3.x.1 $228$ $2$ $2$ $3$ $?$ not computed
228.96.3.x.2 $228$ $2$ $2$ $3$ $?$ not computed
264.96.3.gr.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.gr.2 $264$ $2$ $2$ $3$ $?$ not computed
276.96.3.x.1 $276$ $2$ $2$ $3$ $?$ not computed
276.96.3.x.2 $276$ $2$ $2$ $3$ $?$ not computed
312.96.3.in.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.in.2 $312$ $2$ $2$ $3$ $?$ not computed