| $\GL_2(\Z/10\Z)$-generators: |
$\begin{bmatrix}3&9\\1&4\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: |
$C_{24}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
20.240.3-10.a.1.1, 20.240.3-10.a.1.2, 40.240.3-10.a.1.1, 40.240.3-10.a.1.2, 60.240.3-10.a.1.1, 60.240.3-10.a.1.2, 120.240.3-10.a.1.1, 120.240.3-10.a.1.2, 140.240.3-10.a.1.1, 140.240.3-10.a.1.2, 220.240.3-10.a.1.1, 220.240.3-10.a.1.2, 260.240.3-10.a.1.1, 260.240.3-10.a.1.2, 280.240.3-10.a.1.1, 280.240.3-10.a.1.2 |
| Cyclic 10-isogeny field degree: |
$6$ |
| Cyclic 10-torsion field degree: |
$24$ |
| Full 10-torsion field degree: |
$24$ |
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{4} - x^{3} y - x^{3} z + x^{2} y^{2} + 2 x^{2} y z + x^{2} z^{2} - x y^{3} + 2 x y^{2} z + 2 x y z^{2} + \cdots + z^{4} $ |
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This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 120 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{125x^{3}y^{25}z^{2}-325x^{3}y^{24}z^{3}+425x^{3}y^{23}z^{4}-7250x^{3}y^{22}z^{5}-2625x^{3}y^{21}z^{6}+5200x^{3}y^{20}z^{7}+13975x^{3}y^{19}z^{8}+75725x^{3}y^{18}z^{9}+61000x^{3}y^{17}z^{10}+10500x^{3}y^{16}z^{11}+13425x^{3}y^{15}z^{12}+89025x^{3}y^{14}z^{13}+89025x^{3}y^{13}z^{14}+13425x^{3}y^{12}z^{15}+10500x^{3}y^{11}z^{16}+61000x^{3}y^{10}z^{17}+75725x^{3}y^{9}z^{18}+13975x^{3}y^{8}z^{19}+5200x^{3}y^{7}z^{20}-2625x^{3}y^{6}z^{21}-7250x^{3}y^{5}z^{22}+425x^{3}y^{4}z^{23}-325x^{3}y^{3}z^{24}+125x^{3}y^{2}z^{25}+10x^{2}y^{27}z-80x^{2}y^{26}z^{2}-2175x^{2}y^{24}z^{4}+6000x^{2}y^{23}z^{5}-765x^{2}y^{22}z^{6}+38610x^{2}y^{21}z^{7}+50125x^{2}y^{20}z^{8}-28100x^{2}y^{19}z^{9}-61300x^{2}y^{18}z^{10}-75650x^{2}y^{17}z^{11}-74590x^{2}y^{16}z^{12}-94950x^{2}y^{15}z^{13}-93150x^{2}y^{14}z^{14}-94950x^{2}y^{13}z^{15}-74590x^{2}y^{12}z^{16}-75650x^{2}y^{11}z^{17}-61300x^{2}y^{10}z^{18}-28100x^{2}y^{9}z^{19}+50125x^{2}y^{8}z^{20}+38610x^{2}y^{7}z^{21}-765x^{2}y^{6}z^{22}+6000x^{2}y^{5}z^{23}-2175x^{2}y^{4}z^{24}-80x^{2}y^{2}z^{26}+10x^{2}yz^{27}+10xy^{28}z-35xy^{27}z^{2}+140xy^{26}z^{3}-1200xy^{25}z^{4}-2825xy^{24}z^{5}+11150xy^{23}z^{6}-7315xy^{22}z^{7}+55425xy^{21}z^{8}+24725xy^{20}z^{9}-72900xy^{19}z^{10}-57315xy^{18}z^{11}-6845xy^{17}z^{12}-8810xy^{16}z^{13}-89725xy^{15}z^{14}-89725xy^{14}z^{15}-8810xy^{13}z^{16}-6845xy^{12}z^{17}-57315xy^{11}z^{18}-72900xy^{10}z^{19}+24725xy^{9}z^{20}+55425xy^{8}z^{21}-7315xy^{7}z^{22}+11150xy^{6}z^{23}-2825xy^{5}z^{24}-1200xy^{4}z^{25}+140xy^{3}z^{26}-35xy^{2}z^{27}+10xyz^{28}+y^{30}-10y^{29}z+10y^{28}z^{2}-435y^{27}z^{3}+1185y^{26}z^{4}+1048y^{25}z^{5}+3050y^{24}z^{6}+12325y^{23}z^{7}-23925y^{22}z^{8}-525y^{21}z^{9}-14294y^{20}z^{10}+6965y^{19}z^{11}-13265y^{18}z^{12}-34785y^{17}z^{13}-12090y^{16}z^{14}-16398y^{15}z^{15}-12090y^{14}z^{16}-34785y^{13}z^{17}-13265y^{12}z^{18}+6965y^{11}z^{19}-14294y^{10}z^{20}-525y^{9}z^{21}-23925y^{8}z^{22}+12325y^{7}z^{23}+3050y^{6}z^{24}+1048y^{5}z^{25}+1185y^{4}z^{26}-435y^{3}z^{27}+10y^{2}z^{28}-10yz^{29}+z^{30}}{z^{10}y^{10}(25x^{3}y^{5}z^{2}+125x^{3}y^{4}z^{3}+125x^{3}y^{3}z^{4}+25x^{3}y^{2}z^{5}+10x^{2}y^{7}z-15x^{2}y^{6}z^{2}-100x^{2}y^{5}z^{3}-125x^{2}y^{4}z^{4}-100x^{2}y^{3}z^{5}-15x^{2}y^{2}z^{6}+10x^{2}yz^{7}+35xy^{7}z^{2}-125xy^{5}z^{4}-125xy^{4}z^{5}+35xy^{2}z^{7}+y^{10}-25y^{8}z^{2}-25y^{7}z^{3}+2y^{5}z^{5}-25y^{3}z^{7}-25y^{2}z^{8}+z^{10})}$ |
Hi
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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.
