$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&2\\12&5\end{bmatrix}$, $\begin{bmatrix}11&13\\6&1\end{bmatrix}$, $\begin{bmatrix}13&8\\18&1\end{bmatrix}$, $\begin{bmatrix}19&13\\0&17\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.il.1.1, 24.96.1-24.il.1.2, 24.96.1-24.il.1.3, 24.96.1-24.il.1.4, 24.96.1-24.il.1.5, 24.96.1-24.il.1.6, 24.96.1-24.il.1.7, 24.96.1-24.il.1.8, 120.96.1-24.il.1.1, 120.96.1-24.il.1.2, 120.96.1-24.il.1.3, 120.96.1-24.il.1.4, 120.96.1-24.il.1.5, 120.96.1-24.il.1.6, 120.96.1-24.il.1.7, 120.96.1-24.il.1.8, 168.96.1-24.il.1.1, 168.96.1-24.il.1.2, 168.96.1-24.il.1.3, 168.96.1-24.il.1.4, 168.96.1-24.il.1.5, 168.96.1-24.il.1.6, 168.96.1-24.il.1.7, 168.96.1-24.il.1.8, 264.96.1-24.il.1.1, 264.96.1-24.il.1.2, 264.96.1-24.il.1.3, 264.96.1-24.il.1.4, 264.96.1-24.il.1.5, 264.96.1-24.il.1.6, 264.96.1-24.il.1.7, 264.96.1-24.il.1.8, 312.96.1-24.il.1.1, 312.96.1-24.il.1.2, 312.96.1-24.il.1.3, 312.96.1-24.il.1.4, 312.96.1-24.il.1.5, 312.96.1-24.il.1.6, 312.96.1-24.il.1.7, 312.96.1-24.il.1.8 |
Cyclic 24-isogeny field degree: |
$4$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 y z $ |
| $=$ | $3 x^{2} + 7 y^{2} - 2 y z - y w + z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 28 x^{4} + 2 x^{2} y z + 16 x^{2} z^{2} + y^{2} z^{2} + y z^{3} + z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{7^2}\cdot\frac{9448990857216yz^{11}+1574903079936yz^{10}w-10236058642944yz^{9}w^{2}-2842888534272yz^{8}w^{3}+2762569259136yz^{7}w^{4}+904262885568yz^{6}w^{5}-204328601568yz^{5}w^{6}-86704187184yz^{4}w^{7}+3203128800yz^{3}w^{8}+2866505760yz^{2}w^{9}+54686664yzw^{10}-25019280yw^{11}-1349860517888z^{12}-1574903079936z^{11}w-337805056512z^{10}w^{2}+1343061764352z^{9}w^{3}+1324551125376z^{8}w^{4}+18659654208z^{7}w^{5}-298304772192z^{6}w^{6}-52388950608z^{5}w^{7}+16408031328z^{4}w^{8}+4263770448z^{3}w^{9}-82382832z^{2}w^{10}-85917024zw^{11}-6205977w^{12}}{z^{2}(72yz^{9}+732yz^{8}w+3642yz^{7}w^{2}+12075yz^{6}w^{3}+29946yz^{5}w^{4}+59052yz^{4}w^{5}-1647114yz^{3}w^{6}-1027437yz^{2}w^{7}+57474yzw^{8}+55071yw^{9}-72z^{10}-732z^{9}w-3630z^{8}w^{2}-11951z^{7}w^{3}-29316z^{6}w^{4}-56910z^{5}w^{5}+158011z^{4}w^{6}+291345z^{3}w^{7}+324915z^{2}w^{8}+118387zw^{9}+3078w^{10})}$ |
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.