| $\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&6\\0&17\end{bmatrix}$, $\begin{bmatrix}1&21\\0&5\end{bmatrix}$, $\begin{bmatrix}5&18\\6&11\end{bmatrix}$, $\begin{bmatrix}7&15\\0&19\end{bmatrix}$, $\begin{bmatrix}23&15\\12&23\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
24.288.5-24.y.1.1, 24.288.5-24.y.1.2, 24.288.5-24.y.1.3, 24.288.5-24.y.1.4, 24.288.5-24.y.1.5, 24.288.5-24.y.1.6, 24.288.5-24.y.1.7, 24.288.5-24.y.1.8, 120.288.5-24.y.1.1, 120.288.5-24.y.1.2, 120.288.5-24.y.1.3, 120.288.5-24.y.1.4, 120.288.5-24.y.1.5, 120.288.5-24.y.1.6, 120.288.5-24.y.1.7, 120.288.5-24.y.1.8, 168.288.5-24.y.1.1, 168.288.5-24.y.1.2, 168.288.5-24.y.1.3, 168.288.5-24.y.1.4, 168.288.5-24.y.1.5, 168.288.5-24.y.1.6, 168.288.5-24.y.1.7, 168.288.5-24.y.1.8, 264.288.5-24.y.1.1, 264.288.5-24.y.1.2, 264.288.5-24.y.1.3, 264.288.5-24.y.1.4, 264.288.5-24.y.1.5, 264.288.5-24.y.1.6, 264.288.5-24.y.1.7, 264.288.5-24.y.1.8, 312.288.5-24.y.1.1, 312.288.5-24.y.1.2, 312.288.5-24.y.1.3, 312.288.5-24.y.1.4, 312.288.5-24.y.1.5, 312.288.5-24.y.1.6, 312.288.5-24.y.1.7, 312.288.5-24.y.1.8 |
| Cyclic 24-isogeny field degree: |
$4$ |
| Cyclic 24-torsion field degree: |
$32$ |
| Full 24-torsion field degree: |
$512$ |
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x^{2} + y^{2} + z^{2} + w t $ |
| $=$ | $2 z^{2} + z w - z t - w^{2} - w t - t^{2}$ |
| $=$ | $2 x^{2} - 2 y^{2} - z w + z t + w t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} z^{4} + 24 x^{3} y^{2} z^{3} + 8 x^{3} z^{5} + 48 x^{2} y^{4} z^{2} + 40 x^{2} y^{2} z^{4} + \cdots + 3 z^{8} $ |
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This modular curve has no real points and no $\Q_p$ points for $p=7,31$, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -3^3\,\frac{189zw^{17}+1071zw^{16}t+3528zw^{15}t^{2}+7872zw^{14}t^{3}+13216zw^{13}t^{4}+17160zw^{12}t^{5}+17136zw^{11}t^{6}+12408zw^{10}t^{7}+4698zw^{9}t^{8}-4698zw^{8}t^{9}-12408zw^{7}t^{10}-17136zw^{6}t^{11}-17160zw^{5}t^{12}-13216zw^{4}t^{13}-7872zw^{3}t^{14}-3528zw^{2}t^{15}-1071zwt^{16}-189zt^{17}-99w^{18}-657w^{17}t-2457w^{16}t^{2}-6384w^{15}t^{3}-12614w^{14}t^{4}-19804w^{13}t^{5}-26106w^{12}t^{6}-29712w^{11}t^{7}-30660w^{10}t^{8}-31014w^{9}t^{9}-30660w^{8}t^{10}-29712w^{7}t^{11}-26106w^{6}t^{12}-19804w^{5}t^{13}-12614w^{4}t^{14}-6384w^{3}t^{15}-2457w^{2}t^{16}-657wt^{17}-99t^{18}}{t^{3}w^{3}(w^{2}+wt+t^{2})^{3}(3zw^{5}+5zw^{4}t+6zw^{3}t^{2}-6zw^{2}t^{3}-5zwt^{4}-3zt^{5}+3w^{6}+5w^{5}t+7w^{4}t^{2}+6w^{3}t^{3}+7w^{2}t^{4}+5wt^{5}+3t^{6})}$ |
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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 $X(1)$.
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.
