Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^16 + 1)
gp: K = bnfinit(x^32 - x^16 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
\( x^{32} - x^{16} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(14648040110065267094876580444599852735215435776\)\(\medspace = 2^{128}\cdot 3^{16}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $27.71$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 3$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $32$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(96=2^{5}\cdot 3\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{96}(1,·)$, $\chi_{96}(5,·)$, $\chi_{96}(7,·)$, $\chi_{96}(11,·)$, $\chi_{96}(13,·)$, $\chi_{96}(17,·)$, $\chi_{96}(19,·)$, $\chi_{96}(23,·)$, $\chi_{96}(25,·)$, $\chi_{96}(29,·)$, $\chi_{96}(31,·)$, $\chi_{96}(35,·)$, $\chi_{96}(37,·)$, $\chi_{96}(41,·)$, $\chi_{96}(43,·)$, $\chi_{96}(47,·)$, $\chi_{96}(49,·)$, $\chi_{96}(53,·)$, $\chi_{96}(55,·)$, $\chi_{96}(59,·)$, $\chi_{96}(61,·)$, $\chi_{96}(65,·)$, $\chi_{96}(67,·)$, $\chi_{96}(71,·)$, $\chi_{96}(73,·)$, $\chi_{96}(77,·)$, $\chi_{96}(79,·)$, $\chi_{96}(83,·)$, $\chi_{96}(85,·)$, $\chi_{96}(89,·)$, $\chi_{96}(91,·)$, $\chi_{96}(95,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
$C_{3}\times C_{3}$, which has order $9$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( a \) (order $96$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | \( a^{6} + a^{3} + 1 \), \( a^{18} + a^{9} + 1 \), \( a^{12} + a^{9} + a^{6} + a^{3} + 1 \), \( a^{26} - a^{10} - a^{6} + 1 \), \( a^{26} - a^{10} - a^{6} - 1 \), \( a^{20} - 1 \), \( a^{18} + a^{14} - a^{2} \), \( a^{10} - 1 \), \( a^{5} - 1 \), \( a^{13} - 1 \), \( a^{30} - a^{18} - 1 \), \( a^{18} - a^{2} - 1 \), \( a^{18} - a^{17} + a - 1 \), \( a - 1 \), \( a^{7} + 1 \) (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 47233168152.25636 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2^2\times C_8$ (as 32T37):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^2\times C_8$ |
| Character table for $C_2^2\times C_8$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||