Trait overview

cofactorThe order of the prime order subgroup and its cofactor
discriminantFactorization of the discriminant of the Frobenius polynomial, i.e. factorization of $t^2-4p=v^2d_K$, where $t$ is the trace of Frobenius, $v$ is the maximal conductor and $d_K$ is the CM discriminant.
twist_orderFactorization of the quadratic twist cardinality in an extension, i.e. $\#E(\mathbb{F}_{p^d})$.
kn_factorizationFactorization of $kn \pm 1$ where $n$ is the cardinality of the curve.
torsion_extensionDegrees of field extensions containing the least nontrivial $l$-torsion, the full $l$-torsion and their relative degree of extension.
conductorFactorization of ratio of the maximal conductors of CM-field over an extension and over a basefield.
embeddingThe complement of the embedding degree, i.e. $(n-1)/e$ where $n$ is the prime-subgroup order and $e$ is the embedding degree.
class_numberUpper and lower bound for the class number of the CM-field.
small_prime_orderMultiplicative orders of small primes modulo the prime-subgroup order.
division_polynomialsFactorizations of small division polynomials.
volcanoVolcano depth and crater degree of the $l$-isogeny graph.
isogeny_extensionThe least field extensions containing a nontrivial number and full number of $l$-isogenies and their relative ratio.
trace_factorizationFactorization of trace in field extensions.
isogeny_neighborsNumber of $j$-invariants adjacent to the curve by $l$-isogeny. This is the degree of the point in the $l$-isogeny graph.
q_torsionTorsion order of the lift of $E$ to $Q$.
hamming_xNumber of points with low Hamming weight of the $x$-coordinate and the expected weight.
square_4p1Square parts of $4q \pm 1$ and $4n \pm 1$.
pow_distanceDistance of $n$ from the nearest power of two and multiple of 32/64.
multiples_xBitlength of the $x$-coordinate of small inverted generator scalar multiples, i.e. $x$-coordinate of $P$ where $kP=G$. The difference and ratio to the bitlength of the whole group is also considered.
x962_invariantComputation of $a^3/b^2$.
brainpool_overlapBit overlaps in curve coefficients
weierstrassCoefficients of the curve in Weierstrass form