$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}11&32\\45&41\end{bmatrix}$, $\begin{bmatrix}13&36\\15&7\end{bmatrix}$, $\begin{bmatrix}13&46\\51&11\end{bmatrix}$, $\begin{bmatrix}53&20\\57&59\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.192.3-60.bq.1.1, 60.192.3-60.bq.1.2, 60.192.3-60.bq.1.3, 60.192.3-60.bq.1.4, 60.192.3-60.bq.1.5, 60.192.3-60.bq.1.6, 60.192.3-60.bq.1.7, 60.192.3-60.bq.1.8, 120.192.3-60.bq.1.1, 120.192.3-60.bq.1.2, 120.192.3-60.bq.1.3, 120.192.3-60.bq.1.4, 120.192.3-60.bq.1.5, 120.192.3-60.bq.1.6, 120.192.3-60.bq.1.7, 120.192.3-60.bq.1.8, 120.192.3-60.bq.1.9, 120.192.3-60.bq.1.10, 120.192.3-60.bq.1.11, 120.192.3-60.bq.1.12, 120.192.3-60.bq.1.13, 120.192.3-60.bq.1.14, 120.192.3-60.bq.1.15, 120.192.3-60.bq.1.16, 120.192.3-60.bq.1.17, 120.192.3-60.bq.1.18, 120.192.3-60.bq.1.19, 120.192.3-60.bq.1.20, 120.192.3-60.bq.1.21, 120.192.3-60.bq.1.22, 120.192.3-60.bq.1.23, 120.192.3-60.bq.1.24 |
Cyclic 60-isogeny field degree: |
$12$ |
Cyclic 60-torsion field degree: |
$192$ |
Full 60-torsion field degree: |
$23040$ |
Embedded model Embedded model in $\mathbb{P}^{5}$
$ 0 $ | $=$ | $ x z + y^{2} $ |
| $=$ | $x^{2} - x z + z^{2} - w^{2}$ |
| $=$ | $x y + x z + x u + y z - y w - 2 y t - y u + z u + w u$ |
| $=$ | $x y - x z + x w + x t + x u + y z - y w - y t + y u + z w + z t + z u - w^{2} - w t$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 8 x^{6} y^{2} - 24 x^{5} y^{3} + 120 x^{5} y^{2} z + 52 x^{4} y^{4} - 200 x^{4} y^{3} z + \cdots + 378125 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
$ 4 w^{2} $ | $=$ | $ 12 x^{3} y + 10 x^{2} z^{2} + 20 x y z^{2} + 125 z^{4} $ |
$0$ | $=$ | $2 x^{2} - 2 x y + 2 y^{2} + 5 z^{2}$ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle t$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}u$ |
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 -3^3\cdot5\,\frac{113902500000xt^{11}+1485500000xt^{10}u-7102901050000xt^{9}u^{2}-22250193040000xt^{8}u^{3}-6636505595000xt^{7}u^{4}+39572131987000xt^{6}u^{5}+47183941725500xt^{5}u^{6}+21525457972000xt^{4}u^{7}+4424950047900xt^{3}u^{8}+68206838570xt^{2}u^{9}-154467945265xtu^{10}-25041953844xu^{11}-227500000000yzt^{10}-4550000000000yzt^{9}u-10156868000000yzt^{8}u^{2}+21588880000000yzt^{7}u^{3}+71555668920000yzt^{6}u^{4}+50885641840000yzt^{5}u^{5}+3078488720000yzt^{4}u^{6}-9350969920000yzt^{3}u^{7}-4531838924400yzt^{2}u^{8}-826355940000yztu^{9}-16946008960yzu^{10}+113597500000yt^{11}+4783209500000yt^{10}u+17342798050000yt^{9}u^{2}-8237385860000yt^{8}u^{3}-88343040145000yt^{7}u^{4}-100748454477000yt^{6}u^{5}-34044617003500yt^{5}u^{6}+5878728023000yt^{4}u^{7}+7756281256000yt^{3}u^{8}+2232973599230yt^{2}u^{9}+217559336265ytu^{10}+21624457114yu^{11}-227500000000z^{2}t^{10}+9747368000000z^{2}t^{8}u^{2}+24984688000000z^{2}t^{7}u^{3}+7498277080000z^{2}t^{6}u^{4}-24150717520000z^{2}t^{5}u^{5}-23644652520000z^{2}t^{4}u^{6}-9464294400000z^{2}t^{3}u^{7}-2007773115600z^{2}t^{2}u^{8}-288363632800z^{2}tu^{9}-27640259440z^{2}u^{10}+227500000000zwt^{10}+239020000000zwt^{9}u-5600528000000zwt^{8}u^{2}-16724561600000zwt^{7}u^{3}-21446272680000zwt^{6}u^{4}-15415771720000zwt^{5}u^{5}-1700342880000zwt^{4}u^{6}+3565719760000zwt^{3}u^{7}+1615522835600zwt^{2}u^{8}+191443522000zwtu^{9}+10364601440zwu^{10}+113902500000zt^{11}+1485500000zt^{10}u-7078137050000zt^{9}u^{2}-21555609040000zt^{8}u^{3}-4354877755000zt^{7}u^{4}+39389622387000zt^{6}u^{5}+38434384805500zt^{5}u^{6}+10251847972000zt^{4}u^{7}-1041104860900zt^{3}u^{8}-1124985612630zt^{2}u^{9}-148763435585ztu^{10}+17278079916zu^{11}-113902500000wt^{11}-113163000000wt^{10}u+2572496050000wt^{9}u^{2}+7337252990000wt^{8}u^{3}+15673426715000wt^{7}u^{4}+27667364718000wt^{6}u^{5}+17350530540500wt^{5}u^{6}-2380210266500wt^{4}u^{7}-6072691141500wt^{3}u^{8}-2887393100970wt^{2}u^{9}-701371990355wtu^{10}-65860943961wu^{11}+320000000t^{12}+4224000000t^{11}u-88605900000t^{10}u^{2}-831421820000t^{9}u^{3}-206938250000t^{8}u^{4}+6980410276000t^{7}u^{5}+14859383563000t^{6}u^{6}+11678135209000t^{5}u^{7}+2134315560100t^{4}u^{8}-1911288717440t^{3}u^{9}-1222113653960t^{2}u^{10}-251195774402tu^{11}-3258769331u^{12}}{400000xt^{11}-400000xt^{10}u+6200000xt^{9}u^{2}+58400000xt^{8}u^{3}-319875000xt^{7}u^{4}-1743060000xt^{6}u^{5}+2329960200xt^{5}u^{6}+10549641000xt^{4}u^{7}+8286413575xt^{3}u^{8}+2676950100xt^{2}u^{9}+294729036xtu^{10}+16485164xu^{11}-21600000yzt^{8}u^{2}-144000000yzt^{7}u^{3}-1162390000yzt^{6}u^{4}-2668530000yzt^{5}u^{5}+5933751000yzt^{4}u^{6}+12240806000yzt^{3}u^{7}+4678562650yzt^{2}u^{8}+456685300yztu^{9}+38520960yzu^{10}-400000yt^{11}-400000yt^{10}u+16200000yt^{9}u^{2}+77200000yt^{8}u^{3}+546675000yt^{7}u^{4}-1661380000yt^{6}u^{5}-11099981200yt^{5}u^{6}-6816513000yt^{4}u^{7}+3762227875yt^{3}u^{8}+2915352550yt^{2}u^{9}+659020034ytu^{10}+52172804yu^{11}+21600000z^{2}t^{8}u^{2}+105600000z^{2}t^{7}u^{3}+133840000z^{2}t^{6}u^{4}+2431410000z^{2}t^{5}u^{5}+5491299000z^{2}t^{4}u^{6}+1016706000z^{2}t^{3}u^{7}-1266667150z^{2}t^{2}u^{8}-724697900z^{2}tu^{9}-120003610z^{2}u^{10}+24000000zwt^{9}u+86400000zwt^{8}u^{2}-32000000zwt^{7}u^{3}-581840000zwt^{6}u^{4}-2677820000zwt^{5}u^{5}-4049524000zwt^{4}u^{6}-3775322000zwt^{3}u^{7}-2398267350zwt^{2}u^{8}-844746750zwtu^{9}-98289540zwu^{10}+400000zt^{11}-400000zt^{10}u-37000000zt^{9}u^{2}-200800000zt^{8}u^{3}-849155000zt^{7}u^{4}-1976020000zt^{6}u^{5}+3204262200zt^{5}u^{6}+12373921000zt^{4}u^{7}+9051830375zt^{3}u^{8}+1447115100zt^{2}u^{9}-353537894ztu^{10}-49790296zu^{11}-400000wt^{11}-4000000wt^{10}u-10200000wt^{9}u^{2}-33800000wt^{8}u^{3}+251075000wt^{7}u^{4}+961217000wt^{6}u^{5}-1261311200wt^{5}u^{6}-1630849000wt^{4}u^{7}-1342370575wt^{3}u^{8}-1790562775wt^{2}u^{9}-767547986wtu^{10}-56180950wu^{11}-800000t^{12}+9200000t^{10}u^{2}-400000t^{9}u^{3}-67200000t^{8}u^{4}-46560000t^{7}u^{5}+159266600t^{6}u^{6}+1441983200t^{5}u^{7}+4107626400t^{4}u^{8}+3575373600t^{3}u^{9}+809140233t^{2}u^{10}+36815220tu^{11}-10607340u^{12}}$ |
Hi
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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.