Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} - x w - y^{2} + y w - y t $ |
| $=$ | $x w + x t + y^{2} + y z$ |
| $=$ | $x y + x z + y t - z w - w^{2} + t^{2}$ |
| $=$ | $x y + x z + x w - x t + y z - y t - z^{2} + z w + t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 2 x^{5} y + 2 x^{5} z + x^{4} y^{2} - 2 x^{4} y z - x^{4} z^{2} - 4 x^{3} y^{2} z - 4 x^{3} y z^{2} + \cdots + 3 z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 3x^{4} + x^{2} + 1 $ |
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This modular curve has 7 rational CM points but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve |
CM |
$j$-invariant |
$j$-height | Plane model | Weierstrass model | Embedded model |
| 27.a3 |
$-3$ | $0$ |
| $0.000$ | $(-1/3:4/3:1)$, $(1:0:1)$ | $(-1/3:-28/27:1)$, $(1:0:1)$ | $(-1/4:3/4:0:5/4:1)$, $(-1:-1:0:1:0)$ |
| 121.b1 |
$-11$ | $-32768$ |
$= -1 \cdot 2^{15}$ | $10.397$ | $(0:-1:1)$ | $(0:1:1)$ | $(0:-1:1:0:1)$ |
| 361.a1 |
$-19$ | $-884736$ |
$= -1 \cdot 2^{15} \cdot 3^{3}$ | $13.693$ | $(-1:1:0)$ | $(1:-1:0)$ | $(-1:0:1:-1:1)$ |
| 27.a1 |
$-27$ | $-12288000$ |
$= -1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ | $16.324$ | $(-1:1:0)$ | $(1:1:0)$ | $(-1:0:-3:-1:1)$ |
| 1849.b1 |
$-43$ | $-884736000$ |
$= -1 \cdot 2^{18} \cdot 3^{3} \cdot 5^{3}$ | $20.601$ | $(-1:2:1)$ | $(-1:0:1)$ | $(-1/2:1/2:1:1/2:1)$ |
| 4489.b1 |
$-67$ | $-147197952000$ |
$= -1 \cdot 2^{15} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3}$ | $25.715$ | $(0:3:1)$ | $(0:-1:1)$ | $(0:1/3:-1/3:4/3:1)$ |
| 26569.a1 |
$-163$ | $-262537412640768000$ |
$= -1 \cdot 2^{18} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3} \cdot 29^{3}$ | $40.109$ | $(1/3:-22/3:1)$ | $(1/3:-28/27:1)$ | $(-1/22:-3/22:-7/11:29/22:1)$ |
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x^{2}y+xy^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}x^{7}y^{2}-\frac{3}{2}x^{6}y^{3}+\frac{1}{2}x^{6}y^{2}t+\frac{1}{2}x^{5}y^{4}-3x^{5}y^{3}t+\frac{1}{2}x^{4}y^{5}+\frac{11}{2}x^{4}y^{4}t+\frac{5}{2}x^{3}y^{6}-2x^{3}y^{5}t-\frac{7}{2}x^{2}y^{7}-\frac{7}{2}x^{2}y^{6}t+\frac{1}{2}xy^{8}+3xy^{7}t+\frac{1}{2}y^{9}-\frac{1}{2}y^{8}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -xy^{2}+y^{3}$ |
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 2^9\,\frac{462964514908356xzw^{9}-159622701463004xzw^{8}t-5586499641674960xzw^{7}t^{2}+98339629117519456xzw^{6}t^{3}+660043721859991996xzw^{5}t^{4}+1811818440155005436xzw^{4}t^{5}+2689283905141493832xzw^{3}t^{6}+2169235037419238336xzw^{2}t^{7}+801676094639586933xzwt^{8}+105703773285285725xzt^{9}+764218391014080xw^{10}-1422681489247728xw^{9}t-22522373083574348xw^{8}t^{2}-44221434561617504xw^{7}t^{3}-123507956973093864xw^{6}t^{4}-98944459836447792xw^{5}t^{5}+1297021708803009972xw^{4}t^{6}+3429877626231618864xw^{3}t^{7}+3026222413345985912xw^{2}t^{8}+1105715128834045788xwt^{9}+145001242570506361xt^{10}-344194600999320yw^{10}-613760207953940yw^{9}t+7242416301639188yw^{8}t^{2}+68757669266583168yw^{7}t^{3}+454192207729161240yw^{6}t^{4}+2153115125106462916yw^{5}t^{5}+5521753870957545992yw^{4}t^{6}+7407424001622272512yw^{3}t^{7}+4996199981368079954yw^{2}t^{8}+1580577721310723763ywt^{9}+187394150205078105yt^{10}-878903590461444z^{2}w^{9}-1364411485406936z^{2}w^{8}t+27019692189322672z^{2}w^{7}t^{2}+119136400325639152z^{2}w^{6}t^{3}+498793241211775140z^{2}w^{5}t^{4}+1588895244425821752z^{2}w^{4}t^{5}+2539942294658998408z^{2}w^{3}t^{6}+1869829879661457656z^{2}w^{2}t^{7}+625997847437722899z^{2}wt^{8}+78430539594860914z^{2}t^{9}+882988304923116zw^{10}-797279321277588zw^{9}t-42410312352477976zw^{8}t^{2}-300209294476449984zw^{7}t^{3}-1633225301148501820zw^{6}t^{4}-4912177167195051612zw^{5}t^{5}-7370480269298186708zw^{4}t^{6}-5256849485178408192zw^{3}t^{7}-1578479156802777953zw^{2}t^{8}-99541906717049253zwt^{9}+21478018504394454zt^{10}+420023790014760w^{11}-222007827036788w^{10}t-17171350383279944w^{9}t^{2}-133677282136718828w^{8}t^{3}-804116403857131040w^{7}t^{4}-2622863819236666348w^{6}t^{5}-4175163790899632788w^{5}t^{6}-2809214562478593836w^{4}t^{7}+153164347249758578w^{3}t^{8}+1289628483371624851w^{2}t^{9}+616156135862255766wt^{10}+88556445841800849t^{11}}{160884421854104xzw^{9}+783907179493848xzw^{8}t+7113484196409824xzw^{7}t^{2}+73883443751265056xzw^{6}t^{3}+216253400619214264xzw^{5}t^{4}+513000719280289880xzw^{4}t^{5}+706410178827880560xzw^{3}t^{6}+605995454487672176xzw^{2}t^{7}+290707789582317950xzwt^{8}+58728442699165534xzt^{9}+204716940975016xw^{10}-1041055751788112xw^{9}t-11426620397438432xw^{8}t^{2}-834702714687872xw^{7}t^{3}-72826404598991624xw^{6}t^{4}+125715601244581136xw^{5}t^{5}+431597447303421984xw^{4}t^{6}+821168694343805824xw^{3}t^{7}+803533110384664762xw^{2}t^{8}+405117041183791532xwt^{9}+82944839343229000xt^{10}-99991972555104yw^{10}+52873302391392yw^{9}t+7088960811070772yw^{8}t^{2}+48511681951020768yw^{7}t^{3}+239150957404891664yw^{6}t^{4}+786126101344212080yw^{5}t^{5}+1501610847412189852yw^{4}t^{6}+1888211366852569488yw^{3}t^{7}+1444119582782531728yw^{2}t^{8}+610453274493289792ywt^{9}+106760108225869589yt^{10}-236706127885096z^{2}w^{9}+419472055385488z^{2}w^{8}t+17352685251387808z^{2}w^{7}t^{2}+43497559973750912z^{2}w^{6}t^{3}+233919791207155608z^{2}w^{5}t^{4}+452741967960254512z^{2}w^{4}t^{5}+628676960764944784z^{2}w^{3}t^{6}+520282000819868128z^{2}w^{2}t^{7}+236336555547357662z^{2}wt^{8}+45070283085829636z^{2}t^{9}+190496912111224zw^{10}-2707457285415248zw^{9}t-36990893595416436zw^{8}t^{2}-163100094733990432zw^{7}t^{3}-668512011387826456zw^{6}t^{4}-1357401678464776736zw^{5}t^{5}-1850334192637822364zw^{4}t^{6}-1487566431828487024zw^{3}t^{7}-631922841934763314zw^{2}t^{8}-88224157576129772zwt^{9}+12239936199955595zt^{10}+98750129641384w^{11}-1076167028341984w^{10}t-16508299563052188w^{9}t^{2}-78780150573471152w^{8}t^{3}-342527329534286616w^{7}t^{4}-732986074388433808w^{6}t^{5}-1028158317114886900w^{5}t^{6}-747414337984133392w^{4}t^{7}-96574694801025694w^{3}t^{8}+287309177444371440w^{2}t^{9}+214139649237713681wt^{10}+49433156850459700t^{11}}$ |
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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.
