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Properties

Label 16.0.13573982477...7041.5
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 17^{12}$
Root discriminant $57.32$
Ramified primes $13, 17$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139129, 0, 9180884, 0, 14841596, 0, 7090207, 0, 1343266, 0, 121006, 0, 5456, 0, 119, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 119*x^14 + 5456*x^12 + 121006*x^10 + 1343266*x^8 + 7090207*x^6 + 14841596*x^4 + 9180884*x^2 + 139129)
 
gp: K = bnfinit(x^16 + 119*x^14 + 5456*x^12 + 121006*x^10 + 1343266*x^8 + 7090207*x^6 + 14841596*x^4 + 9180884*x^2 + 139129, 1)
 

Normalized defining polynomial

\( x^{16} + 119 x^{14} + 5456 x^{12} + 121006 x^{10} + 1343266 x^{8} + 7090207 x^{6} + 14841596 x^{4} + 9180884 x^{2} + 139129 \)

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:  \(13573982477229290545823357041=13^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.32$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{11632} a^{12} - \frac{367}{11632} a^{10} - \frac{2447}{11632} a^{8} - \frac{361}{11632} a^{6} + \frac{2903}{11632} a^{4} + \frac{3}{5816} a^{2} - \frac{1}{2} a - \frac{79}{11632}$, $\frac{1}{23264} a^{13} - \frac{1}{23264} a^{12} - \frac{367}{23264} a^{11} + \frac{367}{23264} a^{10} - \frac{2447}{23264} a^{9} + \frac{2447}{23264} a^{8} + \frac{5455}{23264} a^{7} - \frac{5455}{23264} a^{6} - \frac{2913}{23264} a^{5} - \frac{8719}{23264} a^{4} + \frac{2911}{11632} a^{3} - \frac{2911}{11632} a^{2} + \frac{11553}{23264} a + \frac{79}{23264}$, $\frac{1}{1170756606479795648} a^{14} - \frac{3910292541027}{292689151619948912} a^{12} + \frac{28640418123279829}{292689151619948912} a^{10} - \frac{77702130171795527}{585378303239897824} a^{8} + \frac{63652354359199763}{292689151619948912} a^{6} - \frac{347173688083764709}{1170756606479795648} a^{4} - \frac{1}{2} a^{3} - \frac{481951937381030805}{1170756606479795648} a^{2} - \frac{1}{2} a + \frac{631515569491699}{1170756606479795648}$, $\frac{1}{873384428433927553408} a^{15} - \frac{1}{2341513212959591296} a^{14} + \frac{2124268487832451}{109173053554240944176} a^{13} - \frac{10626058665007}{292689151619948912} a^{12} - \frac{4278273684093535001}{109173053554240944176} a^{11} - \frac{9702906850303891}{292689151619948912} a^{10} - \frac{105476343878936847621}{436692214216963776704} a^{9} - \frac{91842187539278731}{1170756606479795648} a^{8} + \frac{18801260315900019573}{109173053554240944176} a^{7} + \frac{45887925707110247}{292689151619948912} a^{6} - \frac{63945768534976797193}{873384428433927553408} a^{5} - \frac{530390518578661207}{2341513212959591296} a^{4} - \frac{159602791684196296637}{873384428433927553408} a^{3} - \frac{689408566935669827}{2341513212959591296} a^{2} - \frac{121931072288602533609}{873384428433927553408} a + \frac{7319805949757257}{2341513212959591296}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( 506567.420659 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

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 $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{17}) \), 4.0.10793861.1, \(\Q(\sqrt{13}, \sqrt{17})\), 4.0.10793861.2, 4.0.2873.1 x2, 4.0.3757.1 x2, 8.0.116507435287321.1, 8.0.2385443281.2, 8.8.116507435287321.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$