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Properties

Label 16.0.43612046166...3616.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{72}\cdot 31^{4}$
Root discriminant $53.39$
Ramified primes $2, 31$
Class number $1184$ (GRH)
Class group $[2, 2, 2, 148]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![222722, -608640, 1259280, -1261120, 958208, -507040, 292784, -138624, 63084, -18864, 9104, -1504, 880, -48, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 48*x^14 - 48*x^13 + 880*x^12 - 1504*x^11 + 9104*x^10 - 18864*x^9 + 63084*x^8 - 138624*x^7 + 292784*x^6 - 507040*x^5 + 958208*x^4 - 1261120*x^3 + 1259280*x^2 - 608640*x + 222722)
 
gp: K = bnfinit(x^16 + 48*x^14 - 48*x^13 + 880*x^12 - 1504*x^11 + 9104*x^10 - 18864*x^9 + 63084*x^8 - 138624*x^7 + 292784*x^6 - 507040*x^5 + 958208*x^4 - 1261120*x^3 + 1259280*x^2 - 608640*x + 222722, 1)
 

Normalized defining polynomial

\( x^{16} + 48 x^{14} - 48 x^{13} + 880 x^{12} - 1504 x^{11} + 9104 x^{10} - 18864 x^{9} + 63084 x^{8} - 138624 x^{7} + 292784 x^{6} - 507040 x^{5} + 958208 x^{4} - 1261120 x^{3} + 1259280 x^{2} - 608640 x + 222722 \)

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:  \(4361204616626257617397743616=2^{72}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.39$
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:  $2, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1386867697955066625189934034253611288526185761} a^{15} + \frac{365232727850996169844545486222810846943347801}{1386867697955066625189934034253611288526185761} a^{14} + \frac{632435817759989358428662185020801601436528027}{1386867697955066625189934034253611288526185761} a^{13} - \frac{540995563710980117460067309153244472611973985}{1386867697955066625189934034253611288526185761} a^{12} + \frac{206808272565642129762090605566480730647271578}{1386867697955066625189934034253611288526185761} a^{11} + \frac{560815999052644949747867479045731635347374147}{1386867697955066625189934034253611288526185761} a^{10} + \frac{517202035508853589429695311529916784669099792}{1386867697955066625189934034253611288526185761} a^{9} + \frac{221169347728993048810170628676418416219161648}{1386867697955066625189934034253611288526185761} a^{8} + \frac{652409903431295787266138793728887273293301172}{1386867697955066625189934034253611288526185761} a^{7} + \frac{460630630060573771950613278240789523299345649}{1386867697955066625189934034253611288526185761} a^{6} + \frac{533953706278791770945276314963443965546419752}{1386867697955066625189934034253611288526185761} a^{5} - \frac{327988581788920428211399784690200321477635120}{1386867697955066625189934034253611288526185761} a^{4} - \frac{210979742679525443665761014125288595676566384}{1386867697955066625189934034253611288526185761} a^{3} - \frac{155110521350601732550336640778138037143274779}{1386867697955066625189934034253611288526185761} a^{2} - \frac{81824652789492775091610013600896459488098349}{1386867697955066625189934034253611288526185761} a - \frac{14707629943962194004797405769920378501575181}{1386867697955066625189934034253611288526185761}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{148}$, which has order $1184$ (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:  \( 15753.9498624 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
31Data not computed