The term "supersingular prime" has two equivalent characterizations: one group-theoretic and the other number-theoretic.
Group-theoretically, let positive integer
and let modular group Gamma0 . Let
compactification (by adding cusps) of
the orbit quotient upper half-plane . The Fricke involution matrix prime number , define iff genus
0 (Ogg 1974-1975, p. 7-01).
Equivalently, iff every supersingular
elliptic curve over the algebraic closure
of the finite field j -invariant
Supersingular curves were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight "
of the television crime drama NUMB3RS
There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (OEIS A002267 ). The supersingular
primes are exactly the set of primes
that divide the group order of the Monster
group .
See also Modular Group Gamma0 ,
Monster Group
This entry contributed by John McKay
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References Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas
of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Conway, J. H. and Norton,
S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11 ,
308-339, 1979. https://doi.org/10.1112/blms/11.3.308 . Ogg,
A. P. "Automorphismes de courbes modulaires." Séminaire
Delange-Pisot-Poitou. Théorie des nombres 16 , 7-01-7-08, 1974-1975.
https://www.numdam.org/article/SDPP_1974-1975__16_1_A4_0.pdf . Ogg,
A. P. "Modular Functions." In The
Santa Cruz Conference on Finite Groups. Held at the University of California, Santa
Cruz, Calif., June 25-July 20, 1979 Silverman,
J. H. The
Arithmetic of Elliptic Curves. Silverman,
J. H. Advanced
Topics in the Arithmetic of Elliptic Curves. Sloane,
N. J. A. Sequence A002267 in "The
On-Line Encyclopedia of Integer Sequences." Referenced on Wolfram|Alpha Supersingular Prime
Cite this as:
Weisstein, Eric W. , with contributions by John McKay . "Supersingular Prime." From MathWorld https://mathworld.wolfram.com/SupersingularPrime.html
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be a positive integer
and let 
be the modular group Gamma0. Let
be the compactification (by adding cusps) of
the orbit quotient 
,
where 
is the upper half-plane. The Fricke involution 
is induced by the matrix ![[[0,-1],[N,0]]](/images/equations/SupersingularPrime/Inline7.svg)
, acting as 
. For 
a prime number, define 
. Then 
is a supersingular prime iff 
has genus
0 (Ogg 1974-1975, p. 7-01).
is a supersingular prime iff every supersingular
elliptic curve over the algebraic closure
of the finite field 
has its j-invariant
in 
(Ogg 1974-1975, p. 7-07).
