| $\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&15\\21&14\end{bmatrix}$, $\begin{bmatrix}7&12\\21&13\end{bmatrix}$, $\begin{bmatrix}11&0\\21&5\end{bmatrix}$, $\begin{bmatrix}13&3\\6&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: |
Group 768.1088670 |
| Contains $-I$: |
yes |
| Quadratic refinements: |
24.192.5-24.dl.1.1, 24.192.5-24.dl.1.2, 24.192.5-24.dl.1.3, 24.192.5-24.dl.1.4, 48.192.5-24.dl.1.1, 48.192.5-24.dl.1.2, 120.192.5-24.dl.1.1, 120.192.5-24.dl.1.2, 120.192.5-24.dl.1.3, 120.192.5-24.dl.1.4, 168.192.5-24.dl.1.1, 168.192.5-24.dl.1.2, 168.192.5-24.dl.1.3, 168.192.5-24.dl.1.4, 240.192.5-24.dl.1.1, 240.192.5-24.dl.1.2, 264.192.5-24.dl.1.1, 264.192.5-24.dl.1.2, 264.192.5-24.dl.1.3, 264.192.5-24.dl.1.4, 312.192.5-24.dl.1.1, 312.192.5-24.dl.1.2, 312.192.5-24.dl.1.3, 312.192.5-24.dl.1.4 |
| Cyclic 24-isogeny field degree: |
$12$ |
| Cyclic 24-torsion field degree: |
$96$ |
| Full 24-torsion field degree: |
$768$ |
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x^{2} - x y - 3 y^{2} - z^{2} - w t $ |
| $=$ | $4 x^{2} - 5 x y + 3 y^{2} - z^{2} - w^{2} - 3 w t - t^{2}$ |
| $=$ | $4 x^{2} + 4 x y + 3 y^{2} + 2 z w - 2 z t - 2 w t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 18 x^{8} - 81 x^{6} y^{2} - 234 x^{6} y z - 189 x^{6} z^{2} - 36 x^{4} y^{4} - 160 x^{4} y^{3} z + \cdots + 72 y^{4} z^{4} $ |
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This modular curve has no $\Q_p$ points for $p=37$, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{3^3}{2^6}\cdot\frac{12873618384192z^{3}w^{9}+51442266849984z^{3}w^{8}t+19065687129856z^{3}w^{7}t^{2}-130656890928896z^{3}w^{6}t^{3}-109834052591744z^{3}w^{5}t^{4}+109834052591744z^{3}w^{4}t^{5}+130656890928896z^{3}w^{3}t^{6}-19065687129856z^{3}w^{2}t^{7}-51442266849984z^{3}wt^{8}-12873618384192z^{3}t^{9}+10127592306504z^{2}w^{10}+28138743139536z^{2}w^{9}t-38359624643928z^{2}w^{8}t^{2}-129093586082880z^{2}w^{7}t^{3}+29040364477968z^{2}w^{6}t^{4}+194697144262368z^{2}w^{5}t^{5}+29040364477968z^{2}w^{4}t^{6}-129093586082880z^{2}w^{3}t^{7}-38359624643928z^{2}w^{2}t^{8}+28138743139536z^{2}wt^{9}+10127592306504z^{2}t^{10}+14448449127960zw^{11}+37795544673336zw^{10}t-18706274746392zw^{9}t^{2}-70923473481336zw^{8}t^{3}+44215670975472zw^{7}t^{4}+67863743616048zw^{6}t^{5}-67863743616048zw^{5}t^{6}-44215670975472zw^{4}t^{7}+70923473481336zw^{3}t^{8}+18706274746392zw^{2}t^{9}-37795544673336zwt^{10}-14448449127960zt^{11}+1723980404997w^{12}-782910818820w^{11}t-14098032352262w^{10}t^{2}-6782855913332w^{9}t^{3}+12218608578283w^{8}t^{4}+2223288849784w^{7}t^{5}-5791789526996w^{6}t^{6}+2223288849784w^{5}t^{7}+12218608578283w^{4}t^{8}-6782855913332w^{3}t^{9}-14098032352262w^{2}t^{10}-782910818820wt^{11}+1723980404997t^{12}}{1620158841z^{3}w^{9}+6990168699z^{3}w^{8}t+15495568804z^{3}w^{7}t^{2}+18390598264z^{3}w^{6}t^{3}+8587274914z^{3}w^{5}t^{4}-8587274914z^{3}w^{4}t^{5}-18390598264z^{3}w^{3}t^{6}-15495568804z^{3}w^{2}t^{7}-6990168699z^{3}wt^{8}-1620158841z^{3}t^{9}+2920983687z^{2}w^{10}+7914914325z^{2}w^{9}t+6027680556z^{2}w^{8}t^{2}-16908954648z^{2}w^{7}t^{3}-52161165120z^{2}w^{6}t^{4}-70592488956z^{2}w^{5}t^{5}-52161165120z^{2}w^{4}t^{6}-16908954648z^{2}w^{3}t^{7}+6027680556z^{2}w^{2}t^{8}+7914914325z^{2}wt^{9}+2920983687z^{2}t^{10}+1443345885zw^{11}+559180773zw^{10}t-10289604738zw^{9}t^{2}-32062339824zw^{8}t^{3}-41665230927zw^{7}t^{4}-20283240531zw^{6}t^{5}+20283240531zw^{5}t^{6}+41665230927zw^{4}t^{7}+32062339824zw^{3}t^{8}+10289604738zw^{2}t^{9}-559180773zwt^{10}-1443345885zt^{11}+142521039w^{12}-365564853w^{11}t-1129619846w^{10}t^{2}-460057940w^{9}t^{3}+4946289955w^{8}t^{4}+12537086359w^{7}t^{5}+16623265417w^{6}t^{6}+12537086359w^{5}t^{7}+4946289955w^{4}t^{8}-460057940w^{3}t^{9}-1129619846w^{2}t^{10}-365564853wt^{11}+142521039t^{12}}$ |
Hi
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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.
