magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^24 + x^16 - x^8 + 1)
gp: K = bnfinit(x^32 - x^24 + x^16 - x^8 + 1, 1)
Normalized defining polynomial
\( x^{32} - x^{24} + x^{16} - x^{8} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(4722366482869645213696000000000000000000000000=2^{96}\cdot 5^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $26.75$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(80=2^{4}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{80}(1,·)$, $\chi_{80}(3,·)$, $\chi_{80}(7,·)$, $\chi_{80}(9,·)$, $\chi_{80}(11,·)$, $\chi_{80}(13,·)$, $\chi_{80}(17,·)$, $\chi_{80}(19,·)$, $\chi_{80}(21,·)$, $\chi_{80}(23,·)$, $\chi_{80}(27,·)$, $\chi_{80}(29,·)$, $\chi_{80}(31,·)$, $\chi_{80}(33,·)$, $\chi_{80}(37,·)$, $\chi_{80}(39,·)$, $\chi_{80}(41,·)$, $\chi_{80}(43,·)$, $\chi_{80}(47,·)$, $\chi_{80}(49,·)$, $\chi_{80}(51,·)$, $\chi_{80}(53,·)$, $\chi_{80}(57,·)$, $\chi_{80}(59,·)$, $\chi_{80}(61,·)$, $\chi_{80}(63,·)$, $\chi_{80}(67,·)$, $\chi_{80}(69,·)$, $\chi_{80}(71,·)$, $\chi_{80}(73,·)$, $\chi_{80}(77,·)$, $\chi_{80}(79,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{5}$, which has order $5$ (assuming GRH)
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( a \) (order $80$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( a^{16} - a^{8} \), \( a^{12} - 1 \), \( a^{29} - a^{25} \), \( a^{6} - 1 \), \( a^{28} - a^{20} - a^{18} + a^{12} - a^{4} \), \( a^{14} - 1 \), \( a^{24} - a^{22} \), \( a^{3} - 1 \), \( a - 1 \), \( a^{9} - 1 \), \( a^{7} - 1 \), \( a^{17} - 1 \), \( a^{11} - 1 \), \( a^{13} - 1 \), \( a^{24} + a^{5} \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 24306760628.406784 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_4^2$ (as 32T36):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_4^2$ |
| Character table for $C_2\times C_4^2$ 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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];
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]])
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 5 | Data not computed | ||||||