| $\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}7&4\\8&19\end{bmatrix}$, $\begin{bmatrix}11&18\\0&5\end{bmatrix}$, $\begin{bmatrix}13&2\\22&13\end{bmatrix}$, $\begin{bmatrix}17&18\\18&1\end{bmatrix}$, $\begin{bmatrix}19&2\\10&1\end{bmatrix}$, $\begin{bmatrix}23&6\\18&17\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
24.72.2-24.c.1.1, 24.72.2-24.c.1.2, 24.72.2-24.c.1.3, 24.72.2-24.c.1.4, 24.72.2-24.c.1.5, 24.72.2-24.c.1.6, 24.72.2-24.c.1.7, 24.72.2-24.c.1.8, 24.72.2-24.c.1.9, 24.72.2-24.c.1.10, 24.72.2-24.c.1.11, 24.72.2-24.c.1.12, 24.72.2-24.c.1.13, 24.72.2-24.c.1.14, 24.72.2-24.c.1.15, 24.72.2-24.c.1.16, 120.72.2-24.c.1.1, 120.72.2-24.c.1.2, 120.72.2-24.c.1.3, 120.72.2-24.c.1.4, 120.72.2-24.c.1.5, 120.72.2-24.c.1.6, 120.72.2-24.c.1.7, 120.72.2-24.c.1.8, 120.72.2-24.c.1.9, 120.72.2-24.c.1.10, 120.72.2-24.c.1.11, 120.72.2-24.c.1.12, 120.72.2-24.c.1.13, 120.72.2-24.c.1.14, 120.72.2-24.c.1.15, 120.72.2-24.c.1.16, 168.72.2-24.c.1.1, 168.72.2-24.c.1.2, 168.72.2-24.c.1.3, 168.72.2-24.c.1.4, 168.72.2-24.c.1.5, 168.72.2-24.c.1.6, 168.72.2-24.c.1.7, 168.72.2-24.c.1.8, 168.72.2-24.c.1.9, 168.72.2-24.c.1.10, 168.72.2-24.c.1.11, 168.72.2-24.c.1.12, 168.72.2-24.c.1.13, 168.72.2-24.c.1.14, 168.72.2-24.c.1.15, 168.72.2-24.c.1.16, 264.72.2-24.c.1.1, 264.72.2-24.c.1.2, 264.72.2-24.c.1.3, 264.72.2-24.c.1.4, 264.72.2-24.c.1.5, 264.72.2-24.c.1.6, 264.72.2-24.c.1.7, 264.72.2-24.c.1.8, 264.72.2-24.c.1.9, 264.72.2-24.c.1.10, 264.72.2-24.c.1.11, 264.72.2-24.c.1.12, 264.72.2-24.c.1.13, 264.72.2-24.c.1.14, 264.72.2-24.c.1.15, 264.72.2-24.c.1.16, 312.72.2-24.c.1.1, 312.72.2-24.c.1.2, 312.72.2-24.c.1.3, 312.72.2-24.c.1.4, 312.72.2-24.c.1.5, 312.72.2-24.c.1.6, 312.72.2-24.c.1.7, 312.72.2-24.c.1.8, 312.72.2-24.c.1.9, 312.72.2-24.c.1.10, 312.72.2-24.c.1.11, 312.72.2-24.c.1.12, 312.72.2-24.c.1.13, 312.72.2-24.c.1.14, 312.72.2-24.c.1.15, 312.72.2-24.c.1.16 |
| Cyclic 24-isogeny field degree: |
$16$ |
| Cyclic 24-torsion field degree: |
$128$ |
| Full 24-torsion field degree: |
$2048$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} y + x z^{2} + y w^{2} $ |
| $=$ | $6 x^{2} w + 24 x y z + w^{3}$ |
| $=$ | $24 y^{2} w - z w^{2}$ |
| $=$ | $24 y^{2} z - z^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{3} y + 27 y^{2} z^{2} + 2 z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + x^{3} y $ | $=$ | $ -54 $ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{18}x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{12}w$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{96}xw^{2}+y^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{12}w$ |
Maps to other modular curves
$j$-invariant map
of degree 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^8\,\frac{216x^{7}w+72x^{5}w^{3}+18x^{3}w^{5}+xw^{7}-64yz^{7}+32yz^{4}w^{3}-8yzw^{6}}{w^{5}(6x^{3}+xw^{2}-4yzw)}$ |
Hi
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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.
This modular curve is minimally covered by the modular curves in the database listed below.
