Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ - x z t + y^{2} t - 2 y z t + y w t - z w t $ |
| $=$ | $ - x z^{2} + y^{2} z - 2 y z^{2} + y z w - z^{2} w$ |
| $=$ | $ - x y z + y^{3} - 2 y^{2} z + y^{2} w - y z w$ |
| $=$ | $ - x z w + y^{2} w - 2 y z w + y w^{2} - z w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{7} - 27 x^{6} y - 18 x^{5} y^{2} - 33 x^{5} z^{2} - 27 x^{4} y^{3} + 81 x^{4} y z^{2} + \cdots + 27 y z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 3x^{7} + 15x^{6} + 21x^{5} + 30x^{4} + 21x^{3} + 15x^{2} + 3x $ |
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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 embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{3251922475xzw^{12}+14069278862xzw^{10}t^{2}-21430210365xzw^{8}t^{4}+4565959665xzw^{6}t^{6}-8097894458xzw^{4}t^{8}+838981983xzw^{2}t^{10}+13107712xzt^{12}+9400624854xw^{11}t^{2}+8107624374xw^{9}t^{4}-32973344316xw^{7}t^{6}+17407412562xw^{5}t^{8}-689032419xw^{3}t^{10}-222386688xwt^{12}-10818593284yzw^{12}-3105752354yzw^{10}t^{2}+17928885897yzw^{8}t^{4}-12269935104yzw^{6}t^{6}+3804983441yzw^{4}t^{8}+638736801yzw^{2}t^{10}+7164928yzt^{12}-1224440063yw^{13}-266746864yw^{11}t^{2}+30061382007yw^{9}t^{4}-23238551577yw^{7}t^{6}+14411146351yw^{5}t^{8}-2972265633yw^{3}t^{10}-151634432ywt^{12}+21637185840z^{2}w^{12}+19092033000z^{2}w^{10}t^{2}-37230778575z^{2}w^{8}t^{4}+10459993068z^{2}w^{6}t^{6}-9600119388z^{2}w^{4}t^{8}+336391965z^{2}w^{2}t^{10}+1024z^{2}t^{12}-803043077zw^{13}-3358735120zw^{11}t^{2}-27884709783zw^{9}t^{4}+14668243953zw^{7}t^{6}-6551384291zw^{5}t^{8}+564916575zw^{3}t^{10}+13103616zwt^{12}+w^{14}-4976390128w^{12}t^{2}-7820880639w^{10}t^{4}-9436550268w^{8}t^{6}+10224277195w^{6}t^{8}+2620713456w^{4}t^{10}-328224863w^{2}t^{12}}{t^{2}(243xzw^{10}+2835xzw^{8}t^{2}-2808xzw^{6}t^{4}-19728xzw^{4}t^{6}-5376xzw^{2}t^{8}+192xzt^{10}-243xw^{9}t^{2}-2592xw^{7}t^{4}+2592xw^{5}t^{6}+13248xw^{3}t^{8}+1920xwt^{10}-243yzw^{10}+567yzw^{8}t^{2}+14256yzw^{6}t^{4}+10656yzw^{4}t^{6}-7296yzw^{2}t^{8}-128yzt^{10}-81yw^{9}t^{2}-3672yw^{7}t^{4}-5616yw^{5}t^{6}+9600yw^{3}t^{8}+3136ywt^{10}+486z^{2}w^{10}+2187z^{2}w^{8}t^{2}-16848z^{2}w^{6}t^{4}-29952z^{2}w^{4}t^{6}+2304z^{2}w^{2}t^{8}+243zw^{9}t^{2}+1080zw^{7}t^{4}-4752zw^{5}t^{6}-4608zw^{3}t^{8}+192zwt^{10}-324w^{10}t^{2}-3429w^{8}t^{4}+6048w^{6}t^{6}+22848w^{4}t^{8}+1408w^{2}t^{10})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.3.gs.4
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 9X^{7}-27X^{6}Y-18X^{5}Y^{2}-27X^{4}Y^{3}+9X^{3}Y^{4}-33X^{5}Z^{2}+81X^{4}YZ^{2}+141X^{3}Y^{2}Z^{2}+9X^{2}Y^{3}Z^{2}+37X^{3}Z^{4}-105X^{2}YZ^{4}-45XY^{2}Z^{4}-9XZ^{6}+27YZ^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.96.3.gs.4
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{13}{1540}z^{7}-\frac{6}{77}z^{6}w+\frac{9}{308}z^{5}w^{2}+\frac{1289}{23100}z^{5}t^{2}+\frac{3}{1540}z^{4}w^{3}+\frac{401}{1925}z^{4}wt^{2}-\frac{603}{3850}z^{3}w^{2}t^{2}-\frac{909}{7700}z^{3}t^{4}+\frac{54}{1925}z^{2}w^{3}t^{2}+\frac{141}{700}z^{2}wt^{4}+\frac{108}{1925}zw^{2}t^{4}+\frac{54}{1925}zt^{6}-\frac{162}{1925}wt^{6}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{941889}{8999178496}z^{27}t-\frac{107475}{1124897312}z^{26}wt+\frac{1555965}{8999178496}z^{25}w^{2}t-\frac{81845333}{44995892480}z^{25}t^{3}-\frac{22173}{2249794624}z^{24}w^{3}t+\frac{14036681}{11248973120}z^{24}wt^{3}-\frac{123793941}{44995892480}z^{23}w^{2}t^{3}+\frac{769703839}{56244865600}z^{23}t^{5}+\frac{97269}{409053568}z^{22}w^{3}t^{3}-\frac{1312556243}{224979462400}z^{22}wt^{5}+\frac{4174130547}{224979462400}z^{21}w^{2}t^{5}-\frac{592739688689}{10124075808000}z^{21}t^{7}-\frac{451140869}{224979462400}z^{20}w^{3}t^{5}+\frac{10068531037}{1265509476000}z^{20}wt^{7}-\frac{116970490669}{1687345968000}z^{19}w^{2}t^{7}+\frac{23897731192837}{151861137120000}z^{19}t^{9}+\frac{191819641}{22598383500}z^{18}w^{3}t^{7}+\frac{4285238839841}{151861137120000}z^{18}wt^{9}+\frac{71163051131}{451967670000}z^{17}w^{2}t^{9}-\frac{1879231599829}{6779515050000}z^{17}t^{11}-\frac{614192111}{30131178000}z^{16}w^{3}t^{9}-\frac{158852067161}{1129919175000}z^{16}wt^{11}-\frac{28195897099}{125546575000}z^{15}w^{2}t^{11}+\frac{3662645955827}{11299191750000}z^{15}t^{13}+\frac{90553954}{3138664375}z^{14}w^{3}t^{11}+\frac{1009489462973}{3766397250000}z^{14}wt^{13}+\frac{15792754471}{78466609375}z^{13}w^{2}t^{13}-\frac{2058927762677}{8238993984375}z^{13}t^{15}-\frac{1868243577}{78466609375}z^{12}w^{3}t^{13}-\frac{760495797716}{2746331328125}z^{12}wt^{15}-\frac{304800741786}{2746331328125}z^{11}w^{2}t^{15}+\frac{1701555810077}{13731656640625}z^{11}t^{17}+\frac{29929579776}{2746331328125}z^{10}w^{3}t^{15}+\frac{89833678593}{549266265625}z^{10}wt^{17}+\frac{99471176496}{2746331328125}z^{9}w^{2}t^{17}-\frac{47041915152}{1248332421875}z^{9}t^{19}-\frac{6977865744}{2746331328125}z^{8}w^{3}t^{17}-\frac{150649334496}{2746331328125}z^{8}wt^{19}-\frac{17558011872}{2746331328125}z^{7}w^{2}t^{19}+\frac{87359528304}{13731656640625}z^{7}t^{21}+\frac{644972544}{2746331328125}z^{6}w^{3}t^{19}+\frac{26667538416}{2746331328125}z^{6}wt^{21}+\frac{1289945088}{2746331328125}z^{5}w^{2}t^{21}-\frac{6250991616}{13731656640625}z^{5}t^{23}-\frac{1934917632}{2746331328125}z^{4}wt^{23}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{137}{1540}z^{7}+\frac{4}{77}z^{6}w-\frac{3}{154}z^{5}w^{2}-\frac{2704}{5775}z^{5}t^{2}-\frac{1}{770}z^{4}w^{3}-\frac{802}{5775}z^{4}wt^{2}+\frac{201}{1925}z^{3}w^{2}t^{2}+\frac{1553}{2310}z^{3}t^{4}-\frac{36}{1925}z^{2}w^{3}t^{2}-\frac{47}{350}z^{2}wt^{4}-\frac{72}{1925}zw^{2}t^{4}-\frac{348}{1925}zt^{6}+\frac{108}{1925}wt^{6}$ |
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.
