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Properties

Label 24.48.1.gk.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $64$
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 $4^{4}\cdot8^{4}$ Cusp orbits $2^{2}\cdot4$
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: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.435

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}11&4\\22&21\end{bmatrix}$, $\begin{bmatrix}11&19\\8&21\end{bmatrix}$, $\begin{bmatrix}23&10\\22&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-24.gk.1.1, 48.96.1-24.gk.1.2, 48.96.1-24.gk.1.3, 48.96.1-24.gk.1.4, 48.96.1-24.gk.1.5, 48.96.1-24.gk.1.6, 240.96.1-24.gk.1.1, 240.96.1-24.gk.1.2, 240.96.1-24.gk.1.3, 240.96.1-24.gk.1.4, 240.96.1-24.gk.1.5, 240.96.1-24.gk.1.6
Cyclic 24-isogeny field degree: $8$
Cyclic 24-torsion field degree: $64$
Full 24-torsion field degree: $1536$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ y^{2} - y z + z^{2} - w^{2} $
$=$ $48 x^{2} + 3 y z - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + x^{2} y^{2} - 12 x^{2} z^{2} + y^{4} - 6 y^{2} z^{2} + 9 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 4x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^4\cdot3^3\,\frac{126yz^{7}w^{4}-252yz^{5}w^{6}+337yz^{3}w^{8}-150yzw^{10}+27z^{12}-108z^{10}w^{2}+162z^{8}w^{4}-154z^{6}w^{6}-92z^{4}w^{8}+165z^{2}w^{10}-125w^{12}}{w^{4}(81yz^{7}-162yz^{5}w^{2}+54yz^{3}w^{4}+12yzw^{6}-27z^{6}w^{2}+81z^{4}w^{4}-54z^{2}w^{6}-w^{8})}$

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
8.24.1.s.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.24.0.cj.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.cw.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.dn.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.du.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.1.bc.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.24.1.bj.1 $24$ $2$ $2$ $1$ $0$ 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
24.144.9.boc.1 $24$ $3$ $3$ $9$ $1$ $1^{8}$
24.192.9.my.1 $24$ $4$ $4$ $9$ $1$ $1^{8}$
48.96.3.mt.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.mu.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
48.96.3.mv.1 $48$ $2$ $2$ $3$ $2$ $1^{2}$
48.96.3.mw.1 $48$ $2$ $2$ $3$ $0$ $1^{2}$
120.240.17.xu.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.yys.1 $120$ $6$ $6$ $17$ $?$ not computed
240.96.3.bmf.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bmg.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bmh.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bmi.1 $240$ $2$ $2$ $3$ $?$ not computed