Properties

Label 16.0.4096000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}$
Root discriminant $9.46$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois Group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -36, 47, -30, -18, 72, -95, 72, -18, -30, 47, -36, 18, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*x + 1)
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*x + 1, 1)

Normalized defining polynomial

\(x^{16} \) \(\mathstrut -\mathstrut 6 x^{15} \) \(\mathstrut +\mathstrut 18 x^{14} \) \(\mathstrut -\mathstrut 36 x^{13} \) \(\mathstrut +\mathstrut 47 x^{12} \) \(\mathstrut -\mathstrut 30 x^{11} \) \(\mathstrut -\mathstrut 18 x^{10} \) \(\mathstrut +\mathstrut 72 x^{9} \) \(\mathstrut -\mathstrut 95 x^{8} \) \(\mathstrut +\mathstrut 72 x^{7} \) \(\mathstrut -\mathstrut 18 x^{6} \) \(\mathstrut -\mathstrut 30 x^{5} \) \(\mathstrut +\mathstrut 47 x^{4} \) \(\mathstrut -\mathstrut 36 x^{3} \) \(\mathstrut +\mathstrut 18 x^{2} \) \(\mathstrut -\mathstrut 6 x \) \(\mathstrut +\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $16$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 8]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(4096000000000000=2^{24}\cdot 5^{12}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $9.46$
magma: Abs(Discriminant(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(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101} a^{14} + \frac{30}{101} a^{13} - \frac{14}{101} a^{12} + \frac{36}{101} a^{11} + \frac{44}{101} a^{10} + \frac{3}{101} a^{9} + \frac{46}{101} a^{8} + \frac{8}{101} a^{7} + \frac{46}{101} a^{6} + \frac{3}{101} a^{5} + \frac{44}{101} a^{4} + \frac{36}{101} a^{3} - \frac{14}{101} a^{2} + \frac{30}{101} a + \frac{1}{101}$, $\frac{1}{101} a^{15} - \frac{5}{101} a^{13} - \frac{49}{101} a^{12} - \frac{26}{101} a^{11} - \frac{4}{101} a^{10} - \frac{44}{101} a^{9} + \frac{42}{101} a^{8} + \frac{8}{101} a^{7} + \frac{37}{101} a^{6} - \frac{46}{101} a^{5} + \frac{29}{101} a^{4} + \frac{17}{101} a^{3} + \frac{46}{101} a^{2} + \frac{10}{101} a - \frac{30}{101}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $7$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( \frac{539}{101} a^{15} - \frac{2902}{101} a^{14} + \frac{7912}{101} a^{13} - \frac{14568}{101} a^{12} + \frac{16551}{101} a^{11} - \frac{6422}{101} a^{10} - \frac{12929}{101} a^{9} + \frac{30243}{101} a^{8} - \frac{32741}{101} a^{7} + \frac{19569}{101} a^{6} + \frac{1042}{101} a^{5} - \frac{14592}{101} a^{4} + \frac{16195}{101} a^{3} - \frac{10025}{101} a^{2} + \frac{4180}{101} a - \frac{993}{101} \) (order $20$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{314}{101} a^{15} - \frac{1707}{101} a^{14} + \frac{4689}{101} a^{13} - \frac{8658}{101} a^{12} + \frac{9871}{101} a^{11} - \frac{3846}{101} a^{10} - \frac{7827}{101} a^{9} + \frac{18193}{101} a^{8} - \frac{19527}{101} a^{7} + \frac{11472}{101} a^{6} + \frac{1039}{101} a^{5} - \frac{9139}{101} a^{4} + \frac{9738}{101} a^{3} - \frac{5694}{101} a^{2} + \frac{2228}{101} a - \frac{421}{101} \),  \( \frac{29}{101} a^{15} - \frac{105}{101} a^{14} + \frac{240}{101} a^{13} - \frac{456}{101} a^{12} + \frac{516}{101} a^{11} - \frac{393}{101} a^{10} + \frac{25}{101} a^{9} + \frac{832}{101} a^{8} - \frac{1416}{101} a^{7} + \frac{1495}{101} a^{6} - \frac{639}{101} a^{5} - \frac{244}{101} a^{4} + \frac{1056}{101} a^{3} - \frac{832}{101} a^{2} + \frac{473}{101} a - \frac{66}{101} \),  \( \frac{204}{101} a^{15} - \frac{1022}{101} a^{14} + \frac{2660}{101} a^{13} - \frac{4778}{101} a^{12} + \frac{5172}{101} a^{11} - \frac{1748}{101} a^{10} - \frac{4164}{101} a^{9} + \frac{9430}{101} a^{8} - \frac{9978}{101} a^{7} + \frac{5986}{101} a^{6} + \frac{377}{101} a^{5} - \frac{4308}{101} a^{4} + \frac{4854}{101} a^{3} - \frac{2972}{101} a^{2} + \frac{1276}{101} a - \frac{274}{101} \),  \( \frac{422}{101} a^{15} - \frac{2377}{101} a^{14} + \frac{6673}{101} a^{13} - \frac{12448}{101} a^{12} + \frac{14455}{101} a^{11} - \frac{5882}{101} a^{10} - \frac{11458}{101} a^{9} + \frac{26855}{101} a^{8} - \frac{29073}{101} a^{7} + 170 a^{6} + \frac{1434}{101} a^{5} - \frac{13671}{101} a^{4} + \frac{14724}{101} a^{3} - \frac{8718}{101} a^{2} + \frac{3408}{101} a - \frac{695}{101} \),  \( \frac{258}{101} a^{15} - \frac{1291}{101} a^{14} + \frac{3309}{101} a^{13} - \frac{5779}{101} a^{12} + \frac{5901}{101} a^{11} - \frac{1175}{101} a^{10} - \frac{6236}{101} a^{9} + \frac{11949}{101} a^{8} - \frac{11496}{101} a^{7} + \frac{6013}{101} a^{6} + \frac{1631}{101} a^{5} - \frac{5690}{101} a^{4} + \frac{5582}{101} a^{3} - \frac{3287}{101} a^{2} + \frac{1523}{101} a - \frac{446}{101} \),  \( \frac{398}{101} a^{15} - \frac{2017}{101} a^{14} + \frac{5170}{101} a^{13} - \frac{8939}{101} a^{12} + \frac{8950}{101} a^{11} - \frac{1258}{101} a^{10} - \frac{10332}{101} a^{9} + \frac{18369}{101} a^{8} - \frac{16487}{101} a^{7} + \frac{7188}{101} a^{6} + \frac{4022}{101} a^{5} - \frac{8930}{101} a^{4} + \frac{7581}{101} a^{3} - \frac{3550}{101} a^{2} + \frac{1343}{101} a - \frac{322}{101} \),  \( \frac{74}{101} a^{15} - \frac{342}{101} a^{14} + \frac{783}{101} a^{13} - \frac{1161}{101} a^{12} + \frac{712}{101} a^{11} + \frac{917}{101} a^{10} - \frac{2565}{101} a^{9} + \frac{2930}{101} a^{8} - \frac{1538}{101} a^{7} - \frac{369}{101} a^{6} + \frac{1832}{101} a^{5} - \frac{1691}{101} a^{4} + \frac{965}{101} a^{3} - \frac{90}{101} a^{2} - \frac{26}{101} a + \frac{64}{101} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 87.2940978199 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 4.2.400.1 x2, 4.0.320.1 x2, 4.0.8000.1 x2, 4.2.2000.1 x2, \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 8.0.2560000.1, 8.0.64000000.3, \(\Q(\zeta_{20})\), 8.4.64000000.1 x2, 8.0.4000000.2 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$