$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&15\\8&23\end{bmatrix}$, $\begin{bmatrix}5&9\\4&19\end{bmatrix}$, $\begin{bmatrix}7&21\\20&7\end{bmatrix}$, $\begin{bmatrix}13&3\\0&13\end{bmatrix}$, $\begin{bmatrix}23&21\\16&17\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
Group 768.1035859 |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.ds.3.1, 24.192.1-24.ds.3.2, 24.192.1-24.ds.3.3, 24.192.1-24.ds.3.4, 24.192.1-24.ds.3.5, 24.192.1-24.ds.3.6, 24.192.1-24.ds.3.7, 24.192.1-24.ds.3.8, 24.192.1-24.ds.3.9, 24.192.1-24.ds.3.10, 24.192.1-24.ds.3.11, 24.192.1-24.ds.3.12, 24.192.1-24.ds.3.13, 24.192.1-24.ds.3.14, 24.192.1-24.ds.3.15, 24.192.1-24.ds.3.16, 120.192.1-24.ds.3.1, 120.192.1-24.ds.3.2, 120.192.1-24.ds.3.3, 120.192.1-24.ds.3.4, 120.192.1-24.ds.3.5, 120.192.1-24.ds.3.6, 120.192.1-24.ds.3.7, 120.192.1-24.ds.3.8, 120.192.1-24.ds.3.9, 120.192.1-24.ds.3.10, 120.192.1-24.ds.3.11, 120.192.1-24.ds.3.12, 120.192.1-24.ds.3.13, 120.192.1-24.ds.3.14, 120.192.1-24.ds.3.15, 120.192.1-24.ds.3.16, 168.192.1-24.ds.3.1, 168.192.1-24.ds.3.2, 168.192.1-24.ds.3.3, 168.192.1-24.ds.3.4, 168.192.1-24.ds.3.5, 168.192.1-24.ds.3.6, 168.192.1-24.ds.3.7, 168.192.1-24.ds.3.8, 168.192.1-24.ds.3.9, 168.192.1-24.ds.3.10, 168.192.1-24.ds.3.11, 168.192.1-24.ds.3.12, 168.192.1-24.ds.3.13, 168.192.1-24.ds.3.14, 168.192.1-24.ds.3.15, 168.192.1-24.ds.3.16, 264.192.1-24.ds.3.1, 264.192.1-24.ds.3.2, 264.192.1-24.ds.3.3, 264.192.1-24.ds.3.4, 264.192.1-24.ds.3.5, 264.192.1-24.ds.3.6, 264.192.1-24.ds.3.7, 264.192.1-24.ds.3.8, 264.192.1-24.ds.3.9, 264.192.1-24.ds.3.10, 264.192.1-24.ds.3.11, 264.192.1-24.ds.3.12, 264.192.1-24.ds.3.13, 264.192.1-24.ds.3.14, 264.192.1-24.ds.3.15, 264.192.1-24.ds.3.16, 312.192.1-24.ds.3.1, 312.192.1-24.ds.3.2, 312.192.1-24.ds.3.3, 312.192.1-24.ds.3.4, 312.192.1-24.ds.3.5, 312.192.1-24.ds.3.6, 312.192.1-24.ds.3.7, 312.192.1-24.ds.3.8, 312.192.1-24.ds.3.9, 312.192.1-24.ds.3.10, 312.192.1-24.ds.3.11, 312.192.1-24.ds.3.12, 312.192.1-24.ds.3.13, 312.192.1-24.ds.3.14, 312.192.1-24.ds.3.15, 312.192.1-24.ds.3.16 |
Cyclic 24-isogeny field degree: |
$2$ |
Cyclic 24-torsion field degree: |
$16$ |
Full 24-torsion field degree: |
$768$ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} + x^{2} + 3x + 3 $ |
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps to other modular curves
$j$-invariant map
of degree 96 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2}\cdot\frac{1456x^{2}y^{30}-7966689408x^{2}y^{28}z^{2}+32076405304320x^{2}y^{26}z^{4}-3158774568329216x^{2}y^{24}z^{6}+74738913625178112x^{2}y^{22}z^{8}-760115779120660480x^{2}y^{20}z^{10}+3640614631552057344x^{2}y^{18}z^{12}-8037686138174963712x^{2}y^{16}z^{14}+19295612392590802944x^{2}y^{14}z^{16}-168702610004777107456x^{2}y^{12}z^{18}+822888455871928467456x^{2}y^{10}z^{20}-988049149294943404032x^{2}y^{8}z^{22}-6244819581607658651648x^{2}y^{6}z^{24}+28540000109292146393088x^{2}y^{4}z^{26}-47709006712645334597632x^{2}y^{2}z^{28}+27262293209257745580032x^{2}z^{30}-737280xy^{30}z+206408540160xy^{28}z^{3}-184624242819072xy^{26}z^{5}+6518171718647808xy^{24}z^{7}-96475355522531328xy^{22}z^{9}+781117757339992064xy^{20}z^{11}-3874043932887220224xy^{18}z^{13}+7766705703430914048xy^{16}z^{15}+50666022130390401024xy^{14}z^{17}-401566426019911434240xy^{12}z^{19}+492426196795829256192xy^{10}z^{21}+5167096418306900361216xy^{8}z^{23}-27475333034143251431424xy^{6}z^{25}+57080428183694195294208xy^{4}z^{27}-47709013256938543120384xy^{2}z^{29}-y^{32}+144948176y^{30}z^{2}-3213491660160y^{28}z^{4}+908309608248320y^{26}z^{6}-29412559729172480y^{24}z^{8}+410736023481417728y^{22}z^{10}-3193478425427312640y^{20}z^{12}+15730316053479161856y^{18}z^{14}-57537753244156035072y^{16}z^{16}+164261323935921995776y^{14}z^{18}-166015275008056623104y^{12}z^{20}-1470571932726388064256y^{10}z^{22}+8546782138996707295232y^{8}z^{24}-19941329691800791678976y^{6}z^{26}+16612318829947608104960y^{4}z^{28}+13631166237508498358272y^{2}z^{30}-27262293490732722290688z^{32}}{y^{2}(y^{2}-8z^{2})(x^{2}y^{26}-344x^{2}y^{24}z^{2}-65024x^{2}y^{22}z^{4}+88956928x^{2}y^{20}z^{6}-21067788288x^{2}y^{18}z^{8}+862905073664x^{2}y^{16}z^{10}-6013476405248x^{2}y^{14}z^{12}-284238972715008x^{2}y^{12}z^{14}+6232344683872256x^{2}y^{10}z^{16}-46862907991916544x^{2}y^{8}z^{18}+138663465827958784x^{2}y^{6}z^{20}-123256009387933696x^{2}y^{4}z^{22}-343597383680x^{2}y^{2}z^{24}-549755813888x^{2}z^{26}+34xy^{26}z-44448xy^{24}z^{3}+19275264xy^{22}z^{5}-3281844224xy^{20}z^{7}+217377710080xy^{18}z^{9}-8888319737856xy^{16}z^{11}+229858814722048xy^{14}z^{13}-3466131603980288xy^{12}z^{15}+28085754664058880xy^{10}z^{17}-109132997814386688xy^{8}z^{19}+154070243663151104xy^{6}z^{21}-103079215104xy^{4}z^{23}-137438953472xy^{2}z^{25}+377y^{26}z^{2}-384168y^{24}z^{4}+133583360y^{22}z^{6}-19616782336y^{20}z^{8}+1131135602688y^{18}z^{10}-33796993875968y^{16}z^{12}+559009973141504y^{14}z^{14}-5000923151597568y^{12}z^{16}+21853313746075648y^{10}z^{18}-31455722940137472y^{8}z^{20}-46220307735773184y^{6}z^{22}+123256730942439424y^{4}z^{24}-893353197568y^{2}z^{26}-1649267441664z^{28})}$ |
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Cover information
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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.