$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}5&3\\6&13\end{bmatrix}$, $\begin{bmatrix}5&21\\14&13\end{bmatrix}$, $\begin{bmatrix}7&21\\20&23\end{bmatrix}$, $\begin{bmatrix}11&18\\0&11\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
Group 768.135405 |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.3-24.fg.1.1, 24.192.3-24.fg.1.2, 24.192.3-24.fg.1.3, 24.192.3-24.fg.1.4, 24.192.3-24.fg.1.5, 24.192.3-24.fg.1.6, 24.192.3-24.fg.1.7, 24.192.3-24.fg.1.8, 120.192.3-24.fg.1.1, 120.192.3-24.fg.1.2, 120.192.3-24.fg.1.3, 120.192.3-24.fg.1.4, 120.192.3-24.fg.1.5, 120.192.3-24.fg.1.6, 120.192.3-24.fg.1.7, 120.192.3-24.fg.1.8, 168.192.3-24.fg.1.1, 168.192.3-24.fg.1.2, 168.192.3-24.fg.1.3, 168.192.3-24.fg.1.4, 168.192.3-24.fg.1.5, 168.192.3-24.fg.1.6, 168.192.3-24.fg.1.7, 168.192.3-24.fg.1.8, 264.192.3-24.fg.1.1, 264.192.3-24.fg.1.2, 264.192.3-24.fg.1.3, 264.192.3-24.fg.1.4, 264.192.3-24.fg.1.5, 264.192.3-24.fg.1.6, 264.192.3-24.fg.1.7, 264.192.3-24.fg.1.8, 312.192.3-24.fg.1.1, 312.192.3-24.fg.1.2, 312.192.3-24.fg.1.3, 312.192.3-24.fg.1.4, 312.192.3-24.fg.1.5, 312.192.3-24.fg.1.6, 312.192.3-24.fg.1.7, 312.192.3-24.fg.1.8 |
Cyclic 24-isogeny field degree: |
$4$ |
Cyclic 24-torsion field degree: |
$32$ |
Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{5}$
$ 0 $ | $=$ | $ x w + y z + z t - 2 z u - 2 w t - w u $ |
| $=$ | $2 x^{2} + x y - x t - 2 y t$ |
| $=$ | $x^{2} - x t + 2 z^{2} + 2 z w + 2 w^{2} + t^{2}$ |
| $=$ | $x^{2} - x t - 2 y z + y u + 2 z^{2} - 2 z t - 2 w^{2} + t^{2} + t u$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{8} + 8 x^{7} y + 16 x^{7} z + 16 x^{6} y^{2} + 28 x^{6} y z + 36 x^{6} z^{2} + 52 x^{5} y^{2} z + \cdots + 52 z^{8} $ |
Weierstrass model Weierstrass model
$ y^{2} + x^{4} y $ | $=$ | $ -7x^{8} - 30x^{4} - 12 $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle u$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Birational map from embedded model to Weierstrass model:
$\displaystyle X$ |
$=$ |
$\displaystyle \frac{13}{20}zw-\frac{1}{5}w^{2}-\frac{3}{8}wu$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{625}z^{8}+\frac{2131}{5000}z^{7}w+\frac{4}{625}z^{7}t+\frac{59323}{20000}z^{6}w^{2}+\frac{62}{625}z^{6}wt-\frac{1851}{10000}z^{6}wu+\frac{346211}{40000}z^{5}w^{3}+\frac{3339}{5000}z^{5}w^{2}t-\frac{4881}{4000}z^{5}w^{2}u+\frac{3}{625}z^{5}wtu+\frac{3714069}{256000}z^{4}w^{4}+\frac{357079}{160000}z^{4}w^{3}t-\frac{4773}{1600}z^{4}w^{3}u+\frac{87}{1250}z^{4}w^{2}tu+\frac{2694277}{160000}z^{3}w^{5}+\frac{344173}{80000}z^{3}w^{4}t-\frac{533793}{160000}z^{3}w^{4}u+\frac{186513}{320000}z^{3}w^{3}tu+\frac{2249507}{160000}z^{2}w^{6}+\frac{246717}{40000}z^{2}w^{5}t-\frac{129459}{80000}z^{2}w^{5}u+\frac{70743}{32000}z^{2}w^{4}tu+\frac{313741}{40000}zw^{7}+\frac{28867}{4000}zw^{6}t+\frac{7293}{40000}zw^{6}u+\frac{295347}{80000}zw^{5}tu+\frac{232281}{80000}w^{8}+\frac{5389}{1250}w^{7}t+\frac{13527}{20000}w^{7}u+\frac{17949}{8000}w^{6}tu$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{5}z^{2}-\frac{23}{40}zw-\frac{19}{20}w^{2}-\frac{3}{20}wu$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 13^3\,\frac{2169213382140xt^{9}u^{2}-3950406334734xt^{8}u^{3}-13144788313728xt^{7}u^{4}-6888262284144xt^{6}u^{5}+1086006953364xt^{5}u^{6}+1387811620554xt^{4}u^{7}+205986081768xt^{3}u^{8}-4768887292xt^{2}u^{9}+501988136xtu^{10}-125497034xu^{11}-3303447244404ywt^{10}-17825273466084ywt^{9}u-2927970807192ywt^{8}u^{2}+32590091555168ywt^{7}u^{3}+24936816072900ywt^{6}u^{4}+2418129363972ywt^{5}u^{5}-2142447021998ywt^{4}u^{6}-417582808984ywt^{3}u^{7}+7529822040ywt^{2}u^{8}-752982204ywtu^{9}+125497034ywu^{10}-1821294356022yt^{11}-13830336596013yt^{10}u-9497583555720yt^{9}u^{2}+26054290907311yt^{8}u^{3}+37600520119858yt^{7}u^{4}+12797747312603yt^{6}u^{5}-2413633477972yt^{5}u^{6}-1981475900969yt^{4}u^{7}-291170366344yt^{3}u^{8}+1254970340yt^{2}u^{9}-5913374006808zwt^{10}+26879560370292zwt^{9}u+67107728080740zwt^{8}u^{2}+14391811163728zwt^{7}u^{3}-33356819071904zwt^{6}u^{4}-18737462357076zwt^{5}u^{5}-1939137085264zwt^{4}u^{6}+189513154060zwt^{3}u^{7}-4643390258zwt^{2}u^{8}+752982204zwtu^{9}-125497034zwu^{10}-4338426764280zt^{11}+13581306431004zt^{10}u+67224677356704zt^{9}u^{2}+51033505123200zt^{8}u^{3}-29898102600920zt^{7}u^{4}-42788813031764zt^{6}u^{5}-12970506307968zt^{5}u^{6}-25011517960zt^{4}u^{7}+342976344704zt^{3}u^{8}+752982204zt^{2}u^{9}+3437563097934w^{2}t^{10}+49957350936900w^{2}t^{9}u+44361784723896w^{2}t^{8}u^{2}-49318986868576w^{2}t^{7}u^{3}-61540341823254w^{2}t^{6}u^{4}-15096073174260w^{2}t^{5}u^{5}+2072758559934w^{2}t^{4}u^{6}+765579035048w^{2}t^{3}u^{7}-8157307210w^{2}t^{2}u^{8}+752982204w^{2}tu^{9}-125497034w^{2}u^{10}-2632413745776wt^{11}+21125266526298wt^{10}u+60267675875676wt^{9}u^{2}-4797170516132wt^{8}u^{3}-91203968661688wt^{7}u^{4}-53969726882998wt^{6}u^{5}-276405493442wt^{5}u^{6}+5763867217556wt^{4}u^{7}+997915589128wt^{3}u^{8}-10290756788wt^{2}u^{9}+878479238wtu^{10}-125497034wu^{11}-2078312281654t^{12}-13830336596013t^{11}u-7328370173580t^{10}u^{2}+19263637691809t^{9}u^{3}+9063295870738t^{8}u^{4}-8365231298101t^{7}u^{5}+2359946312952t^{6}u^{6}+6993544961233t^{5}u^{7}+2207931108392t^{4}u^{8}+213857877040t^{3}u^{9}-1505964408t^{2}u^{10}-125497034tu^{11}}{53566402256xt^{9}u^{2}-328245594746xt^{8}u^{3}-1758044143168xt^{7}u^{4}+1609405191854xt^{6}u^{5}+3843869307456xt^{5}u^{6}+620057253728xt^{4}u^{7}-714865199888xt^{3}u^{8}-306416596736xt^{2}u^{9}-50140013440xtu^{10}-3593888672xu^{11}-35679044858ywt^{10}-909755576480ywt^{9}u+1814349275628ywt^{8}u^{2}+8118140458192ywt^{7}u^{3}-1658740076686ywt^{6}u^{4}-10508749760640ywt^{5}u^{5}-3806346900320ywt^{4}u^{6}+687404150528ywt^{3}u^{7}+544348713456ywt^{2}u^{8}+101550532480ywtu^{9}+7185504608ywu^{10}-48945567300yt^{11}-500861283767yt^{10}u+389683469283yt^{9}u^{2}+6977186200007yt^{8}u^{3}+3295229713034yt^{7}u^{4}-10385848077544yt^{6}u^{5}-8619442689616yt^{5}u^{6}-599940983008yt^{4}u^{7}+1257378482336yt^{3}u^{8}+475256324412yt^{2}u^{9}+71024148720ytu^{10}+4635510464yu^{11}-225215395368zwt^{10}+1647963010206zwt^{9}u+6449069498042zwt^{8}u^{2}-12201887173648zwt^{7}u^{3}-20388633451874zwt^{6}u^{4}+4624888207584zwt^{5}u^{5}+10219438931104zwt^{4}u^{6}+2969604871424zwt^{3}u^{7}+150804967208zwt^{2}u^{8}-50723558304zwtu^{9}-5998715360zwu^{10}-107132804512zt^{11}+654389311804zt^{10}u+6171235866088zt^{9}u^{2}-4351420020756zt^{8}u^{3}-26011385709184zt^{7}u^{4}-7387313149632zt^{6}u^{5}+14853575865856zt^{5}u^{6}+9414034961664zt^{4}u^{7}+1378304421888zt^{3}u^{8}-245795339584zt^{2}u^{9}-91221323264ztu^{10}-8084231680zu^{11}-37783078748w^{2}t^{10}+2934799434504w^{2}t^{9}u+256308382630w^{2}t^{8}u^{2}-23585093847248w^{2}t^{7}u^{3}-7776252721682w^{2}t^{6}u^{4}+21414730005888w^{2}t^{5}u^{5}+12289360332384w^{2}t^{4}u^{6}+648836076544w^{2}t^{3}u^{7}-785332126928w^{2}t^{2}u^{8}-192771855744w^{2}tu^{9}-15269736288w^{2}u^{10}-90296385958wt^{11}+1003603210666wt^{10}u+4416740770310wt^{9}u^{2}-11671369594564wt^{8}u^{3}-24073170047070wt^{7}u^{4}+11971355767646wt^{6}u^{5}+27014052975552wt^{5}u^{6}+6637809390624wt^{4}u^{7}-2591216877264wt^{3}u^{8}-1395103738472wt^{2}u^{9}-236413325312wtu^{10}-16456525536wu^{11}-48945567300t^{12}-500861283767t^{11}u+443249871539t^{10}u^{2}+6649991544105t^{9}u^{3}+338906950674t^{8}u^{4}-8249736684602t^{7}u^{5}+1492568409456t^{6}u^{6}+2188817316928t^{5}u^{7}-2178526551408t^{4}u^{8}-1632217006148t^{3}u^{9}-373809826064t^{2}u^{10}-33839081760tu^{11}-1636152784u^{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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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.