| $\GL_2(\Z/12\Z)$-generators: |
$\begin{bmatrix}4&9\\9&11\end{bmatrix}$, $\begin{bmatrix}5&5\\9&8\end{bmatrix}$, $\begin{bmatrix}11&8\\0&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: |
$C_6\times D_6$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
12.128.1-12.a.2.1, 12.128.1-12.a.2.2, 12.128.1-12.a.2.3, 12.128.1-12.a.2.4, 24.128.1-12.a.2.1, 24.128.1-12.a.2.2, 24.128.1-12.a.2.3, 24.128.1-12.a.2.4, 60.128.1-12.a.2.1, 60.128.1-12.a.2.2, 60.128.1-12.a.2.3, 60.128.1-12.a.2.4, 84.128.1-12.a.2.1, 84.128.1-12.a.2.2, 84.128.1-12.a.2.3, 84.128.1-12.a.2.4, 120.128.1-12.a.2.1, 120.128.1-12.a.2.2, 120.128.1-12.a.2.3, 120.128.1-12.a.2.4, 132.128.1-12.a.2.1, 132.128.1-12.a.2.2, 132.128.1-12.a.2.3, 132.128.1-12.a.2.4, 156.128.1-12.a.2.1, 156.128.1-12.a.2.2, 156.128.1-12.a.2.3, 156.128.1-12.a.2.4, 168.128.1-12.a.2.1, 168.128.1-12.a.2.2, 168.128.1-12.a.2.3, 168.128.1-12.a.2.4, 204.128.1-12.a.2.1, 204.128.1-12.a.2.2, 204.128.1-12.a.2.3, 204.128.1-12.a.2.4, 228.128.1-12.a.2.1, 228.128.1-12.a.2.2, 228.128.1-12.a.2.3, 228.128.1-12.a.2.4, 264.128.1-12.a.2.1, 264.128.1-12.a.2.2, 264.128.1-12.a.2.3, 264.128.1-12.a.2.4, 276.128.1-12.a.2.1, 276.128.1-12.a.2.2, 276.128.1-12.a.2.3, 276.128.1-12.a.2.4, 312.128.1-12.a.2.1, 312.128.1-12.a.2.2, 312.128.1-12.a.2.3, 312.128.1-12.a.2.4 |
| Cyclic 12-isogeny field degree: |
$6$ |
| Cyclic 12-torsion field degree: |
$24$ |
| Full 12-torsion field degree: |
$72$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} + 3 y^{2} - z^{2} - 2 z w - w^{2} $ |
| $=$ | $x^{2} + 4 x y + 2 x z + y^{2} - 2 y z - 3 z^{2} - 2 z w - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 795 x^{4} + 312 x^{3} y + 876 x^{3} z + 6 x^{2} y^{2} + 48 x^{2} y z + 738 x^{2} z^{2} + 24 x y^{3} + \cdots + 75 z^{4} $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 3y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 64 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{223400595142002469932xz^{15}+762272922110143268328xz^{14}w+328415607936972175080xz^{13}w^{2}-2184233534212695196704xz^{12}w^{3}-5040009606406048218516xz^{11}w^{4}-5580299963065674006600xz^{10}w^{5}-3742786217733723756816xz^{9}w^{6}-1542393559909845464976xz^{8}w^{7}-315540574029680543676xz^{7}w^{8}+36530522335572111000xz^{6}w^{9}+50150693353271776584xz^{5}w^{10}+18520921213859475840xz^{4}w^{11}+4099359737847336996xz^{3}w^{12}+588905142585292104xz^{2}w^{13}+51613953822852960xzw^{14}+2125884169505712xw^{15}-455409812338824030417y^{2}z^{14}-2341002590161983152064y^{2}z^{13}w-3951429559116287488020y^{2}z^{12}w^{2}-2407912269334765151904y^{2}z^{11}w^{3}+646001880883198722531y^{2}z^{10}w^{4}+1847838559740464642076y^{2}z^{9}w^{5}+1151255775199900267854y^{2}z^{8}w^{6}+281238749504651171376y^{2}z^{7}w^{7}-45623888539735553883y^{2}z^{6}w^{8}-58130190106459749432y^{2}z^{5}w^{9}-21219466523176171800y^{2}z^{4}w^{10}-4459714740484415280y^{2}z^{3}w^{11}-578596715224535799y^{2}z^{2}w^{12}-43251243722936964y^{2}zw^{13}-1372877549767746y^{2}w^{14}+25183361262446447460yz^{15}-1356743098870787455632yz^{14}w-4765670803849113545256yz^{13}w^{2}-6124813755935026881432yz^{12}w^{3}-2870363614978328686668yz^{11}w^{4}+1327328209145658832992yz^{10}w^{5}+2742999497531735904000yz^{9}w^{6}+1877348979169049350008yz^{8}w^{7}+743341663803974210316yz^{7}w^{8}+175565731493682063600yz^{6}w^{9}+17501507693421226200yz^{5}w^{10}-3353845820287931496yz^{4}w^{11}-1661871195035915556yz^{3}w^{12}-312600321646721856yz^{2}w^{13}-31305223109801520yzw^{14}-1402780262898168yw^{15}+442173073723013882810z^{16}+1145180959524158415062z^{15}w+198568650224177569779z^{14}w^{2}-2530516698556772105840z^{13}w^{3}-4271662329616539793001z^{12}w^{4}-3518054885701836521382z^{11}w^{5}-1552729796676675012055z^{10}w^{6}-103530615908413031536z^{9}w^{7}+362283057934549862121z^{8}w^{8}+284642268625051713002z^{7}w^{9}+122307507919491693557z^{6}w^{10}+34497185371351972368z^{5}w^{11}+6560027017932899137z^{4}w^{12}+824682340262641894z^{3}w^{13}+65675063001524559z^{2}w^{14}+3213903748977680zw^{15}+93060575879909w^{16}}{643705058254481304xz^{15}+3358486472607455352xz^{14}w+8608120616315060796xz^{13}w^{2}+14277994234616473944xz^{12}w^{3}+17085165776439676872xz^{11}w^{4}+15595530706329065184xz^{10}w^{5}+11188842833064233700xz^{9}w^{6}+6401370505500409032xz^{8}w^{7}+2932966912116985128xz^{7}w^{8}+1072945529775162384xz^{6}w^{9}+310292239556951916xz^{5}w^{10}+69625303589519832xz^{4}w^{11}+11733679254732072xz^{3}w^{12}+1408249369327416xz^{2}w^{13}+109424475979104xzw^{14}+4200810478128xw^{15}-576956759566933332y^{2}z^{14}-3233331434365641180y^{2}z^{13}w-8068001346376790049y^{2}z^{12}w^{2}-12562388713963916784y^{2}z^{11}w^{3}-13754584247191074540y^{2}z^{10}w^{4}-11232481255691442120y^{2}z^{9}w^{5}-7053538445447329935y^{2}z^{8}w^{6}-3458993660033978088y^{2}z^{7}w^{7}-1330223115327024600y^{2}z^{6}w^{8}-398882354976886572y^{2}z^{5}w^{9}-91644286328493129y^{2}z^{4}w^{10}-15604122066648588y^{2}z^{3}w^{11}-1849403400969366y^{2}z^{2}w^{12}-132413923868688y^{2}zw^{13}-3984736726080y^{2}w^{14}+496208961480245688yz^{15}+862965072439892640yz^{14}w-474294372545981340yz^{13}w^{2}-3762879109400487912yz^{12}w^{3}-6929226439429804488yz^{11}w^{4}-7827403108362209064yz^{10}w^{5}-6324605661950128212yz^{9}w^{6}-3875148847675413792yz^{8}w^{7}-1847894285621041416yz^{7}w^{8}-692389581197619096yz^{6}w^{9}-203413676356023804yz^{5}w^{10}-46215950701480560yz^{4}w^{11}-7890070878324888yz^{3}w^{12}-958281095370456yz^{2}w^{13}-75112797496176yzw^{14}-2996503310328yw^{15}+991906815833523260z^{16}+4382562370318339532z^{15}w+10003965693218807379z^{14}w^{2}+15402026444395950898z^{13}w^{3}+17754956321554602799z^{12}w^{4}+16126193208670891368z^{11}w^{5}+11844652291456811321z^{10}w^{6}+7121414799386846618z^{9}w^{7}+3516186983201165493z^{8}w^{8}+1420875051791476292z^{7}w^{9}+465344204370888227z^{6}w^{10}+121469191242147870z^{5}w^{11}+24612959174610301z^{4}w^{12}+3711632542180288z^{3}w^{13}+385379919988998z^{2}w^{14}+23453667734024zw^{15}+586797693152w^{16}}$ |
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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.
