$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&18\\0&19\end{bmatrix}$, $\begin{bmatrix}7&6\\8&1\end{bmatrix}$, $\begin{bmatrix}15&8\\16&19\end{bmatrix}$, $\begin{bmatrix}17&16\\0&5\end{bmatrix}$, $\begin{bmatrix}21&10\\4&21\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
24.96.1-24.s.2.1, 24.96.1-24.s.2.2, 24.96.1-24.s.2.3, 24.96.1-24.s.2.4, 24.96.1-24.s.2.5, 24.96.1-24.s.2.6, 24.96.1-24.s.2.7, 24.96.1-24.s.2.8, 24.96.1-24.s.2.9, 24.96.1-24.s.2.10, 24.96.1-24.s.2.11, 24.96.1-24.s.2.12, 48.96.1-24.s.2.1, 48.96.1-24.s.2.2, 48.96.1-24.s.2.3, 48.96.1-24.s.2.4, 48.96.1-24.s.2.5, 48.96.1-24.s.2.6, 48.96.1-24.s.2.7, 48.96.1-24.s.2.8, 120.96.1-24.s.2.1, 120.96.1-24.s.2.2, 120.96.1-24.s.2.3, 120.96.1-24.s.2.4, 120.96.1-24.s.2.5, 120.96.1-24.s.2.6, 120.96.1-24.s.2.7, 120.96.1-24.s.2.8, 120.96.1-24.s.2.9, 120.96.1-24.s.2.10, 120.96.1-24.s.2.11, 120.96.1-24.s.2.12, 168.96.1-24.s.2.1, 168.96.1-24.s.2.2, 168.96.1-24.s.2.3, 168.96.1-24.s.2.4, 168.96.1-24.s.2.5, 168.96.1-24.s.2.6, 168.96.1-24.s.2.7, 168.96.1-24.s.2.8, 168.96.1-24.s.2.9, 168.96.1-24.s.2.10, 168.96.1-24.s.2.11, 168.96.1-24.s.2.12, 240.96.1-24.s.2.1, 240.96.1-24.s.2.2, 240.96.1-24.s.2.3, 240.96.1-24.s.2.4, 240.96.1-24.s.2.5, 240.96.1-24.s.2.6, 240.96.1-24.s.2.7, 240.96.1-24.s.2.8, 264.96.1-24.s.2.1, 264.96.1-24.s.2.2, 264.96.1-24.s.2.3, 264.96.1-24.s.2.4, 264.96.1-24.s.2.5, 264.96.1-24.s.2.6, 264.96.1-24.s.2.7, 264.96.1-24.s.2.8, 264.96.1-24.s.2.9, 264.96.1-24.s.2.10, 264.96.1-24.s.2.11, 264.96.1-24.s.2.12, 312.96.1-24.s.2.1, 312.96.1-24.s.2.2, 312.96.1-24.s.2.3, 312.96.1-24.s.2.4, 312.96.1-24.s.2.5, 312.96.1-24.s.2.6, 312.96.1-24.s.2.7, 312.96.1-24.s.2.8, 312.96.1-24.s.2.9, 312.96.1-24.s.2.10, 312.96.1-24.s.2.11, 312.96.1-24.s.2.12 |
Cyclic 24-isogeny field degree: |
$8$ |
Cyclic 24-torsion field degree: |
$64$ |
Full 24-torsion field degree: |
$1536$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x y + 3 x z - y^{2} + 2 y z - z^{2} $ |
| $=$ | $x^{2} - x y - x z + x w + 3 y^{2} + y z + y w + 3 z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 25 x^{4} + 4 x^{3} y + 59 x^{3} z + x^{2} y^{2} + 8 x^{2} y z + 84 x^{2} z^{2} + 2 x y^{2} z + \cdots + 25 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
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 -3\,\frac{122933828878536xz^{11}+13175005494582xz^{10}w+138321488819304xz^{9}w^{2}+358227029415243xz^{8}w^{3}-226899606633264xz^{7}w^{4}-159804811788552xz^{6}w^{5}+475257177464376xz^{5}w^{6}+130088993203422xz^{4}w^{7}+354300629134824xz^{3}w^{8}-217557954724452xz^{2}w^{9}-266481021054848y^{2}z^{10}+318152792646560y^{2}z^{9}w-592133669979165y^{2}z^{8}w^{2}+417728137200080y^{2}z^{7}w^{3}+625608966898820y^{2}z^{6}w^{4}-390704544422328y^{2}z^{5}w^{5}-625608966898820y^{2}z^{4}w^{6}+417728137200080y^{2}z^{3}w^{7}+592133669979165y^{2}z^{2}w^{8}+318152792646560y^{2}zw^{9}+266481021054848y^{2}w^{10}-143547192176312yz^{11}+331327798141142yz^{10}w-236254226435409yz^{9}w^{2}+1130255795750147yz^{8}w^{3}+268620367062134yz^{7}w^{4}-75252178746504yz^{6}w^{5}+9453022354108yz^{5}w^{6}+320917523770238yz^{4}w^{7}+588207269698746yz^{3}w^{8}+238916326741412yz^{2}w^{9}+253306015560266yzw^{10}+122933828878536yw^{11}-48179760535687z^{12}+331327798141142z^{11}w+42646250910291z^{10}w^{2}+804030686295947z^{9}w^{3}+554965550134409z^{8}w^{4}-31782106147104z^{7}w^{5}-135453185640642z^{6}w^{6}+277447451170838z^{5}w^{7}+874552452771021z^{4}w^{8}+565141436195612z^{3}w^{9}+532206492905966z^{2}w^{10}+122933828878536zw^{11}+95367431640625w^{12}}{6993997164xz^{11}-42128296182xz^{10}w-79263189504xz^{9}w^{2}-422368148643xz^{8}w^{3}+598333486464xz^{7}w^{4}+913955761152xz^{6}w^{5}-84781138176xz^{5}w^{6}-152384758272xz^{4}w^{7}+72075241476xz^{3}w^{8}+9325329552xz^{2}w^{9}+2602805373y^{2}z^{10}+91143257940y^{2}z^{9}w+1799483040y^{2}z^{8}w^{2}-218848814080y^{2}z^{7}w^{3}-1957522012320y^{2}z^{6}w^{4}+46800793728y^{2}z^{5}w^{5}+1957522012320y^{2}z^{4}w^{6}-218848814080y^{2}z^{3}w^{7}-1799483040y^{2}z^{2}w^{8}+91143257940y^{2}zw^{9}-2602805373y^{2}w^{10}+9596802537yz^{11}+49014961758yz^{10}w-86789036016yz^{9}w^{2}-569141721247yz^{8}w^{3}-1206803767584yz^{7}w^{4}+875975416704yz^{6}w^{5}+958785112992yz^{5}w^{6}+227099914112yz^{4}w^{7}+492643907079yz^{3}w^{8}+21205397988yz^{2}w^{9}+39525490809yzw^{10}+6993997164yw^{11}+9596802537z^{12}+49014961758z^{11}w-74860901091z^{10}w^{2}-537639932047z^{9}w^{3}-1474694937984z^{8}w^{4}+476460722304z^{7}w^{5}+959435188992z^{6}w^{6}+626614608512z^{5}w^{7}+224752736679z^{4}w^{8}-10296391212z^{3}w^{9}+51453625734z^{2}w^{10}+6993997164zw^{11}}$ |
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.