Curve name | $X_{207f}$ | |||||||||||||||
Index | $96$ | |||||||||||||||
Level | $32$ | |||||||||||||||
Genus | $0$ | |||||||||||||||
Does the subgroup contain $-I$? | No | |||||||||||||||
Generating matrices | $ \left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 9 & 27 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 24 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right]$ | |||||||||||||||
Images in lower levels |
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Meaning/Special name | ||||||||||||||||
Chosen covering | $X_{207}$ | |||||||||||||||
Curves that $X_{207f}$ minimally covers | ||||||||||||||||
Curves that minimally cover $X_{207f}$ | ||||||||||||||||
Curves that minimally cover $X_{207f}$ and have infinitely many rational points. | ||||||||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{26} - 1188t^{25} + 2052t^{24} + 749088t^{23} + 15657408t^{22} + 159314688t^{21} + 871395840t^{20} + 1614034944t^{19} - 11619237888t^{18} - 99396550656t^{17} - 310057058304t^{16} - 58470432768t^{15} + 3189650227200t^{14} + 12170626596864t^{13} + 16041212116992t^{12} - 29041310564352t^{11} - 167626584096768t^{10} - 308513473560576t^{9} - 138365122904064t^{8} + 586416044113920t^{7} + 1516375784816640t^{6} + 1836523720802304t^{5} + 1251252822343680t^{4} + 409121404747776t^{3} - 11132555231232t^{2} - 44530220924928t - 7421703487488\] \[B(t) = -54t^{39} - 3564t^{38} - 219672t^{37} - 8900496t^{36} - 211522752t^{35} - 3030172416t^{34} - 24928929792t^{33} - 64059393024t^{32} + 1081127264256t^{31} + 15765190410240t^{30} + 105924913201152t^{29} + 368185278726144t^{28} - 33037641842688t^{27} - 7621814859595776t^{26} - 40763486141153280t^{25} - 86770633421094912t^{24} + 143513501174857728t^{23} + 1615856026572029952t^{22} + 4873796325482692608t^{21} + 3154413829937430528t^{20} - 28818268225787658240t^{19} - 117515609958437093376t^{18} - 179551947234868199424t^{17} + 143042733745612259328t^{16} + 1316088864829744349184t^{15} + 2905389137340716285952t^{14} + 2324805007525923520512t^{13} - 4149880060480352944128t^{12} - 16531820293939336839168t^{11} - 26925203069642458791936t^{10} - 24185538582160814899200t^{9} - 6321564311251423592448t^{8} + 14476115062940869066752t^{7} + 23627276383370386341888t^{6} + 18893832171317316550656t^{5} + 9319330234115158966272t^{4} + 2876016736437308227584t^{3} + 542809855887711141888t^{2} + 70039981404865953792t + 7782220156096217088\] | |||||||||||||||
Info about rational points | ||||||||||||||||
Comments on finding rational points | None | |||||||||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 + xy = x^3 + x^2 + 31750x - 98631000$, with conductor $1050$ | |||||||||||||||
Generic density of odd order reductions | $1091/10752$ |