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Properties

Label 16.0.61976445190...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 5^{14}\cdot 31^{4}$
Root discriminant $54.58$
Ramified primes $2, 5, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times D_4).C_2^3$ (as 16T315)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23088025, 0, 17874600, 0, 2738850, 0, -934650, 0, -25140, 0, 10020, 0, -75, 0, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 30*x^14 - 75*x^12 + 10020*x^10 - 25140*x^8 - 934650*x^6 + 2738850*x^4 + 17874600*x^2 + 23088025)
 
gp: K = bnfinit(x^16 - 30*x^14 - 75*x^12 + 10020*x^10 - 25140*x^8 - 934650*x^6 + 2738850*x^4 + 17874600*x^2 + 23088025, 1)
 

Normalized defining polynomial

\( x^{16} - 30 x^{14} - 75 x^{12} + 10020 x^{10} - 25140 x^{8} - 934650 x^{6} + 2738850 x^{4} + 17874600 x^{2} + 23088025 \)

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:  \(6197644519014400000000000000=2^{40}\cdot 5^{14}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.58$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8}$, $\frac{1}{5} a^{9}$, $\frac{1}{155} a^{10} + \frac{1}{155} a^{8} - \frac{15}{31} a^{6} - \frac{11}{31} a^{4} - \frac{6}{31} a^{2}$, $\frac{1}{155} a^{11} + \frac{1}{155} a^{9} - \frac{15}{31} a^{7} - \frac{11}{31} a^{5} - \frac{6}{31} a^{3}$, $\frac{1}{24025} a^{12} - \frac{6}{4805} a^{10} - \frac{3}{961} a^{8} + \frac{82}{4805} a^{6} - \frac{1184}{4805} a^{4} + \frac{3}{31} a^{2}$, $\frac{1}{24025} a^{13} - \frac{6}{4805} a^{11} - \frac{3}{961} a^{9} + \frac{82}{4805} a^{7} - \frac{1184}{4805} a^{5} + \frac{3}{31} a^{3}$, $\frac{1}{372352307156362044025} a^{14} + \frac{4904611253697022}{372352307156362044025} a^{12} - \frac{47923858369789720}{14894092286254481761} a^{10} - \frac{4274704827373823507}{74470461431272408805} a^{8} - \frac{2869797072882771261}{14894092286254481761} a^{6} - \frac{548878454338963878}{2402272949395884155} a^{4} + \frac{1203974958488697}{15498535157392801} a^{2} + \frac{221636029108697}{499952747012671}$, $\frac{1}{372352307156362044025} a^{15} + \frac{4904611253697022}{372352307156362044025} a^{13} - \frac{47923858369789720}{14894092286254481761} a^{11} - \frac{4274704827373823507}{74470461431272408805} a^{9} - \frac{2869797072882771261}{14894092286254481761} a^{7} - \frac{548878454338963878}{2402272949395884155} a^{5} + \frac{1203974958488697}{15498535157392801} a^{3} + \frac{221636029108697}{499952747012671} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{12593781824846}{74470461431272408805} a^{14} + \frac{324189319718181}{74470461431272408805} a^{12} + \frac{1999050274942764}{74470461431272408805} a^{10} - \frac{22739498819196626}{14894092286254481761} a^{8} + \frac{5046829962161590}{14894092286254481761} a^{6} + \frac{55397726797084938}{480454589879176831} a^{4} - \frac{4867163388551240}{15498535157392801} a^{2} - \frac{35374193948816}{499952747012671} \) (order $10$)
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:  \( 35251994.9545 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4).C_2^3$ (as 16T315):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times D_4).C_2^3$
Character table for $(C_2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), 4.0.8000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$