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Properties

Label 16.0.34881684083...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 13^{4}\cdot 29^{8}$
Root discriminant $34.19$
Ramified primes $5, 13, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45, 255, 410, 245, 271, -147, 1555, -1523, 2108, -1527, 1255, -590, 302, -84, 29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 29*x^14 - 84*x^13 + 302*x^12 - 590*x^11 + 1255*x^10 - 1527*x^9 + 2108*x^8 - 1523*x^7 + 1555*x^6 - 147*x^5 + 271*x^4 + 245*x^3 + 410*x^2 + 255*x + 45)
 
gp: K = bnfinit(x^16 - 4*x^15 + 29*x^14 - 84*x^13 + 302*x^12 - 590*x^11 + 1255*x^10 - 1527*x^9 + 2108*x^8 - 1523*x^7 + 1555*x^6 - 147*x^5 + 271*x^4 + 245*x^3 + 410*x^2 + 255*x + 45, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 29 x^{14} - 84 x^{13} + 302 x^{12} - 590 x^{11} + 1255 x^{10} - 1527 x^{9} + 2108 x^{8} - 1523 x^{7} + 1555 x^{6} - 147 x^{5} + 271 x^{4} + 245 x^{3} + 410 x^{2} + 255 x + 45 \)

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:  \(3488168408344511962890625=5^{12}\cdot 13^{4}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.19$
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:  $5, 13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3}$, $\frac{1}{36} a^{12} - \frac{1}{12} a^{11} + \frac{1}{36} a^{10} - \frac{1}{36} a^{9} + \frac{1}{6} a^{8} + \frac{5}{12} a^{7} + \frac{1}{4} a^{5} + \frac{2}{9} a^{4} - \frac{1}{12} a^{3} + \frac{11}{36} a^{2} - \frac{7}{18} a - \frac{5}{12}$, $\frac{1}{36} a^{13} - \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{36} a^{5} + \frac{1}{12} a^{4} + \frac{7}{18} a^{3} + \frac{1}{36} a^{2} - \frac{5}{12} a - \frac{1}{4}$, $\frac{1}{36} a^{14} + \frac{1}{18} a^{11} - \frac{1}{36} a^{10} + \frac{1}{36} a^{9} + \frac{1}{12} a^{8} - \frac{5}{12} a^{7} - \frac{1}{36} a^{6} - \frac{5}{12} a^{5} + \frac{1}{3} a^{4} + \frac{7}{36} a^{3} - \frac{11}{36} a^{2} - \frac{7}{36} a + \frac{1}{6}$, $\frac{1}{652046360277132} a^{15} - \frac{5994072087413}{652046360277132} a^{14} - \frac{439269938167}{652046360277132} a^{13} + \frac{3598234073489}{326023180138566} a^{12} + \frac{13558942289777}{217348786759044} a^{11} - \frac{536301629212}{54337196689761} a^{10} - \frac{1797702416839}{652046360277132} a^{9} - \frac{44352621435383}{217348786759044} a^{8} + \frac{242754696385673}{652046360277132} a^{7} - \frac{273492376452073}{652046360277132} a^{6} - \frac{62397122557934}{163011590069283} a^{5} + \frac{14268205867363}{326023180138566} a^{4} - \frac{19356452680189}{108674393379522} a^{3} + \frac{12563115647981}{217348786759044} a^{2} - \frac{80699348060948}{163011590069283} a - \frac{35085463413593}{217348786759044}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 141395.158359 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.3625.1 x2, 4.0.105125.1 x2, 4.0.10933.1, \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.273325.1, 8.0.74706555625.2, 8.8.1867663890625.1, 8.0.11051265625.4

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$