| $\GL_2(\Z/12\Z)$-generators: |
$\begin{bmatrix}5&8\\3&11\end{bmatrix}$, $\begin{bmatrix}7&1\\0&1\end{bmatrix}$, $\begin{bmatrix}11&8\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: |
$S_3\times D_6$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
12.128.1-12.b.1.1, 12.128.1-12.b.1.2, 12.128.1-12.b.1.3, 12.128.1-12.b.1.4, 24.128.1-12.b.1.1, 24.128.1-12.b.1.2, 24.128.1-12.b.1.3, 24.128.1-12.b.1.4, 60.128.1-12.b.1.1, 60.128.1-12.b.1.2, 60.128.1-12.b.1.3, 60.128.1-12.b.1.4, 84.128.1-12.b.1.1, 84.128.1-12.b.1.2, 84.128.1-12.b.1.3, 84.128.1-12.b.1.4, 120.128.1-12.b.1.1, 120.128.1-12.b.1.2, 120.128.1-12.b.1.3, 120.128.1-12.b.1.4, 132.128.1-12.b.1.1, 132.128.1-12.b.1.2, 132.128.1-12.b.1.3, 132.128.1-12.b.1.4, 156.128.1-12.b.1.1, 156.128.1-12.b.1.2, 156.128.1-12.b.1.3, 156.128.1-12.b.1.4, 168.128.1-12.b.1.1, 168.128.1-12.b.1.2, 168.128.1-12.b.1.3, 168.128.1-12.b.1.4, 204.128.1-12.b.1.1, 204.128.1-12.b.1.2, 204.128.1-12.b.1.3, 204.128.1-12.b.1.4, 228.128.1-12.b.1.1, 228.128.1-12.b.1.2, 228.128.1-12.b.1.3, 228.128.1-12.b.1.4, 264.128.1-12.b.1.1, 264.128.1-12.b.1.2, 264.128.1-12.b.1.3, 264.128.1-12.b.1.4, 276.128.1-12.b.1.1, 276.128.1-12.b.1.2, 276.128.1-12.b.1.3, 276.128.1-12.b.1.4, 312.128.1-12.b.1.1, 312.128.1-12.b.1.2, 312.128.1-12.b.1.3, 312.128.1-12.b.1.4 |
| Cyclic 12-isogeny field degree: |
$3$ |
| Cyclic 12-torsion field degree: |
$6$ |
| Full 12-torsion field degree: |
$72$ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 4x - 4 $ |
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This modular curve has rational points, including 2 rational_cusps and 2 known non-cuspidal non-CM points. The following are the known rational points on this modular curve (one row per $j$-invariant).
Maps to other modular curves
$j$-invariant map
of degree 64 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^6}\cdot\frac{12x^{2}y^{25}+353x^{2}y^{24}z-2160x^{2}y^{23}z^{2}-27684x^{2}y^{22}z^{3}+257856x^{2}y^{21}z^{4}-1023246x^{2}y^{20}z^{5}-9678960x^{2}y^{19}z^{6}+42962880x^{2}y^{18}z^{7}-4375541736x^{2}y^{17}z^{8}-16802015874x^{2}y^{16}z^{9}-171372821440x^{2}y^{15}z^{10}-3102643517004x^{2}y^{14}z^{11}-5773771295856x^{2}y^{13}z^{12}-78900540944454x^{2}y^{12}z^{13}-16781498538768x^{2}y^{11}z^{14}+877558708384368x^{2}y^{10}z^{15}+2712976205147140x^{2}y^{9}z^{16}+24876492121885671x^{2}y^{8}z^{17}+5050660482721488x^{2}y^{7}z^{18}-106063152071449564x^{2}y^{6}z^{19}-321360990956391456x^{2}y^{5}z^{20}-2283212045828428593x^{2}y^{4}z^{21}-92901037616256040x^{2}y^{3}z^{22}+4360574882411809488x^{2}y^{2}z^{23}+10494414075786312300x^{2}yz^{24}+65828602281104885579x^{2}z^{25}-xy^{26}-32xy^{25}z-492xy^{24}z^{2}-2676xy^{23}z^{3}+29424xy^{22}z^{4}+1238736xy^{21}z^{5}-13349004xy^{20}z^{6}-142897848xy^{19}z^{7}+507433374xy^{18}z^{8}-20903826688xy^{17}z^{9}-147967736624xy^{16}z^{10}-762474493956xy^{15}z^{11}-14330849293260xy^{14}z^{12}-22176100145616xy^{13}z^{13}-235306741786260xy^{12}z^{14}+91064169474856xy^{11}z^{15}+4469093405700149xy^{10}z^{16}+9453931910579104xy^{9}z^{17}+74159082004319780xy^{8}z^{18}-7635577897747580xy^{7}z^{19}-503359615632479304xy^{6}z^{20}-1014320572963837264xy^{5}z^{21}-6752926090451429906xy^{4}z^{22}+595819578604863964xy^{3}z^{23}+18567440213949420263xy^{2}z^{24}+31483245525435678720xyz^{25}+197485807943544733696xz^{26}-39y^{26}z-184y^{25}z^{2}+7927y^{24}z^{3}+21420y^{23}z^{4}-1034940y^{22}z^{5}+2390136y^{21}z^{6}+21639438y^{20}z^{7}-941196360y^{19}z^{8}-1479238878y^{18}z^{9}-57260687232y^{17}z^{10}-723091429981y^{16}z^{11}-1942234208948y^{15}z^{12}-34985653717032y^{14}z^{13}-32032314085080y^{13}z^{14}-36779637937074y^{12}z^{15}+829504821908472y^{11}z^{16}+11467224354195859y^{10}z^{17}+10591059186611784y^{9}z^{18}+33512209432732005y^{8}z^{19}-111518884461015292y^{7}z^{20}-1152183546689071244y^{6}z^{21}-821100812345118756y^{5}z^{22}-3625713495687204727y^{4}z^{23}+4186859745997491908y^{3}z^{24}+36149733117615653337y^{2}z^{25}+20988831449191224912yz^{26}+131657205663158297300z^{27}}{z^{4}(6x^{2}y^{20}z+1386x^{2}y^{18}z^{3}-21054x^{2}y^{16}z^{5}-15117648x^{2}y^{14}z^{7}+311723896x^{2}y^{12}z^{9}+46148527062x^{2}y^{10}z^{11}+1291398238206x^{2}y^{8}z^{13}+16914654955080x^{2}y^{6}z^{15}+117326695370382x^{2}y^{4}z^{17}+417470821171686x^{2}y^{2}z^{19}+601157982484341x^{2}z^{21}+xy^{22}-144xy^{20}z^{2}+14339xy^{18}z^{4}-658212xy^{16}z^{6}-55181468xy^{14}z^{8}+2596430048xy^{12}z^{10}+205532081019xy^{10}z^{12}+4931025108228xy^{8}z^{14}+59037815143269xy^{6}z^{16}+384334074740736xy^{4}z^{18}+1302508962054873xy^{2}z^{20}+1803473947459584xz^{22}-7y^{22}z+387y^{20}z^{3}+25757y^{18}z^{5}-4043544y^{16}z^{7}-37750860y^{14}z^{9}+12147225373y^{12}z^{11}+491572435905y^{10}z^{13}+8444353102956y^{8}z^{15}+77791789837125y^{6}z^{17}+400598042133717y^{4}z^{19}+1085424135042447y^{2}z^{21}+1202315964981804z^{23})}$ |
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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.
