$\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&6\\0&23\end{bmatrix}$, $\begin{bmatrix}11&6\\0&13\end{bmatrix}$, $\begin{bmatrix}19&3\\20&23\end{bmatrix}$, $\begin{bmatrix}19&6\\8&23\end{bmatrix}$, $\begin{bmatrix}19&6\\12&1\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: |
Group 768.1035912 |
Contains $-I$: |
yes |
Quadratic refinements: |
24.192.1-24.da.1.1, 24.192.1-24.da.1.2, 24.192.1-24.da.1.3, 24.192.1-24.da.1.4, 24.192.1-24.da.1.5, 24.192.1-24.da.1.6, 24.192.1-24.da.1.7, 24.192.1-24.da.1.8, 24.192.1-24.da.1.9, 24.192.1-24.da.1.10, 24.192.1-24.da.1.11, 24.192.1-24.da.1.12, 24.192.1-24.da.1.13, 24.192.1-24.da.1.14, 24.192.1-24.da.1.15, 24.192.1-24.da.1.16, 120.192.1-24.da.1.1, 120.192.1-24.da.1.2, 120.192.1-24.da.1.3, 120.192.1-24.da.1.4, 120.192.1-24.da.1.5, 120.192.1-24.da.1.6, 120.192.1-24.da.1.7, 120.192.1-24.da.1.8, 120.192.1-24.da.1.9, 120.192.1-24.da.1.10, 120.192.1-24.da.1.11, 120.192.1-24.da.1.12, 120.192.1-24.da.1.13, 120.192.1-24.da.1.14, 120.192.1-24.da.1.15, 120.192.1-24.da.1.16, 168.192.1-24.da.1.1, 168.192.1-24.da.1.2, 168.192.1-24.da.1.3, 168.192.1-24.da.1.4, 168.192.1-24.da.1.5, 168.192.1-24.da.1.6, 168.192.1-24.da.1.7, 168.192.1-24.da.1.8, 168.192.1-24.da.1.9, 168.192.1-24.da.1.10, 168.192.1-24.da.1.11, 168.192.1-24.da.1.12, 168.192.1-24.da.1.13, 168.192.1-24.da.1.14, 168.192.1-24.da.1.15, 168.192.1-24.da.1.16, 264.192.1-24.da.1.1, 264.192.1-24.da.1.2, 264.192.1-24.da.1.3, 264.192.1-24.da.1.4, 264.192.1-24.da.1.5, 264.192.1-24.da.1.6, 264.192.1-24.da.1.7, 264.192.1-24.da.1.8, 264.192.1-24.da.1.9, 264.192.1-24.da.1.10, 264.192.1-24.da.1.11, 264.192.1-24.da.1.12, 264.192.1-24.da.1.13, 264.192.1-24.da.1.14, 264.192.1-24.da.1.15, 264.192.1-24.da.1.16, 312.192.1-24.da.1.1, 312.192.1-24.da.1.2, 312.192.1-24.da.1.3, 312.192.1-24.da.1.4, 312.192.1-24.da.1.5, 312.192.1-24.da.1.6, 312.192.1-24.da.1.7, 312.192.1-24.da.1.8, 312.192.1-24.da.1.9, 312.192.1-24.da.1.10, 312.192.1-24.da.1.11, 312.192.1-24.da.1.12, 312.192.1-24.da.1.13, 312.192.1-24.da.1.14, 312.192.1-24.da.1.15, 312.192.1-24.da.1.16 |
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} + 24x + 56 $ |
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
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}{2^3\cdot3^3}\cdot\frac{48x^{2}y^{30}+5692032x^{2}y^{28}z^{2}+188620876800x^{2}y^{26}z^{4}+865293713842176x^{2}y^{24}z^{6}-18848487595015471104x^{2}y^{22}z^{8}-40618300162102009528320x^{2}y^{20}z^{10}+10067995772968125887152128x^{2}y^{18}z^{12}+15473670617226567981289439232x^{2}y^{16}z^{14}-794061797776012679744248086528x^{2}y^{14}z^{16}-445561858290011775724810200416256x^{2}y^{12}z^{18}+16553690172472716947879918926036992x^{2}y^{10}z^{20}+1282889542337633135313051365274550272x^{2}y^{8}z^{22}-22416156065344262934735628130838380544x^{2}y^{6}z^{24}-207500837287607204438376295614426120192x^{2}y^{4}z^{26}-29405065906854270953324094403897196544x^{2}y^{2}z^{28}+454034544748725490855096936058918535168x^{2}z^{30}-96xy^{30}z+44603136xy^{28}z^{3}+4318642096128xy^{26}z^{5}+95349934306934784xy^{24}z^{7}+264834223411475644416xy^{22}z^{9}-202168256505657135464448xy^{20}z^{11}-326010707038073564713451520xy^{18}z^{13}+28318858415885770964286308352xy^{16}z^{15}+23314170523492828896304627187712xy^{14}z^{17}-1254067647065260649666920870576128xy^{12}z^{19}-175640356042420846165764222172004352xy^{10}z^{21}+6280089793223554685578833408942931968xy^{8}z^{23}+109078772270106141130118994521481019392xy^{6}z^{25}-647561431005250698698444672932503355392xy^{4}z^{27}-3060667747674787596117320625848536006656xy^{2}z^{29}-908069089497450981710193872117837070336xz^{31}-y^{32}-176640y^{30}z^{2}-10764845568y^{28}z^{4}-235955358056448y^{26}z^{6}-903725526741073920y^{24}z^{8}+2161786961867009163264y^{22}z^{10}+3746998723354151425671168y^{20}z^{12}-546350747573344709011046400y^{18}z^{14}-490000357235068143306471899136y^{16}z^{16}+44766027472498693035127197401088y^{14}z^{18}+4906717146574462988126524211724288y^{12}z^{20}-456290103371475960071254237771726848y^{10}z^{22}+2046651966672012326530636387253747712y^{8}z^{24}+303155130874927691644962365807693660160y^{6}z^{26}-383483689011572341941729617474513534976y^{4}z^{28}-5925548112296553353585485694613464285184y^{2}z^{30}-3654728615697158484080862699595140956160z^{32}}{z^{2}y^{2}(y^{2}-216z^{2})^{6}(36x^{2}y^{14}-150336x^{2}y^{12}z^{2}+320806656x^{2}y^{10}z^{4}-593535983616x^{2}y^{8}z^{6}+129901662683136x^{2}y^{6}z^{8}-10438106657587200x^{2}y^{4}z^{10}+296148833645101056x^{2}y^{2}z^{12}+252xy^{14}z+1226016xy^{12}z^{3}-2238928128xy^{10}z^{5}-2081366710272xy^{8}z^{7}+826245624840192xy^{6}z^{9}-114858668772163584xy^{4}z^{11}+7403720841127526400xy^{2}z^{13}-191904444202025484288xz^{15}-y^{16}-11520y^{14}z^{2}+41824512y^{12}z^{4}-60089942016y^{10}z^{6}+1252375437312y^{8}z^{8}+2255216157130752y^{6}z^{10}-336140887693197312y^{4}z^{12}+18953525353286467584y^{2}z^{14}-383808888404050968576z^{16})}$ |
Hi
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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.