| $\text{cofactor}()$ |
The order of the prime order subgroup and its cofactor |
| $\text{discriminant}()$ |
Factorization 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. |
| $\text{twist_order}(deg)$ |
Factorization of the quadratic twist cardinality in an extension, i.e. $\#E(\mathbb{F}_{p^d})$. |
| $\text{kn_factorization}(k)$ |
Factorization of $kn \pm 1$ where $n$ is the cardinality of the curve. |
| $\text{torsion_extension}(l)$ |
Degrees of field extensions containing the least nontrivial $l$-torsion, the full $l$-torsion and their relative degree of extension. |
| $\text{conductor}(deg)$ |
Factorization of ratio of the maximal conductors of CM-field over an extension and over a basefield. |
| $\text{embedding}()$ |
The complement of the embedding degree, i.e. $(n-1)/e$ where $n$ is the prime-subgroup order and $e$ is the embedding degree. |
| $\text{class_number}()$ |
Upper and lower bound for the class number of the CM-field. |
| $\text{small_prime_order}(l)$ |
Multiplicative orders of small primes modulo the prime-subgroup order. |
| $\text{division_polynomials}(l)$ |
Factorizations of small division polynomials. |
| $\text{volcano}(l)$ |
Volcano depth and crater degree of the $l$-isogeny graph. |
| $\text{isogeny_extension}(l)$ |
The least field extensions containing a nontrivial number and full number of $l$-isogenies and their relative ratio. |
| $\text{trace_factorization}(deg)$ |
Factorization of trace in field extensions. |
| $\text{isogeny_neighbors}(l)$ |
Number of $j$-invariants adjacent to the curve by $l$-isogeny. This is the degree of the point in the $l$-isogeny graph. |
| $\text{q_torsion}()$ |
Torsion order of the lift of $E$ to $Q$. |
| $\text{hamming_x}(weight)$ |
Number of points with low Hamming weight of the $x$-coordinate and the expected weight. |
| $\text{square_4p1}()$ |
Square parts of $4q \pm 1$ and $4n \pm 1$. |
| $\text{pow_distance}()$ |
Distance of $n$ from the nearest power of two and multiple of 32/64. |
| $\text{multiples_x}(k)$ |
Bitlength 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. |
| $\text{x962_invariant}()$ |
Computation of $a^3/b^2$. |
| $\text{brainpool_overlap}()$ |
Bit overlaps in curve coefficients |
| $\text{weierstrass}()$ |
Coefficients of the curve in Weierstrass form |