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Properties

Label 16.8.35093559222...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 3^{8}\cdot 5^{12}\cdot 11^{2}\cdot 29^{4}$
Root discriminant $25.65$
Ramified primes $2, 3, 5, 11, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -18, -154, -302, -66, 398, 247, 159, -49, -170, -133, 29, 64, 10, -15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 15*x^14 + 10*x^13 + 64*x^12 + 29*x^11 - 133*x^10 - 170*x^9 - 49*x^8 + 159*x^7 + 247*x^6 + 398*x^5 - 66*x^4 - 302*x^3 - 154*x^2 - 18*x + 31)
 
gp: K = bnfinit(x^16 - x^15 - 15*x^14 + 10*x^13 + 64*x^12 + 29*x^11 - 133*x^10 - 170*x^9 - 49*x^8 + 159*x^7 + 247*x^6 + 398*x^5 - 66*x^4 - 302*x^3 - 154*x^2 - 18*x + 31, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 15 x^{14} + 10 x^{13} + 64 x^{12} + 29 x^{11} - 133 x^{10} - 170 x^{9} - 49 x^{8} + 159 x^{7} + 247 x^{6} + 398 x^{5} - 66 x^{4} - 302 x^{3} - 154 x^{2} - 18 x + 31 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35093559222562500000000=2^{8}\cdot 3^{8}\cdot 5^{12}\cdot 11^{2}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.65$
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:  $2, 3, 5, 11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{22} a^{13} + \frac{1}{11} a^{12} - \frac{3}{22} a^{11} - \frac{1}{2} a^{10} - \frac{1}{11} a^{9} - \frac{2}{11} a^{8} - \frac{3}{22} a^{7} + \frac{5}{11} a^{6} + \frac{2}{11} a^{5} + \frac{3}{11} a^{4} + \frac{5}{22} a^{3} + \frac{1}{22} a^{2} - \frac{1}{22} a - \frac{1}{22}$, $\frac{1}{22} a^{14} + \frac{2}{11} a^{12} + \frac{3}{11} a^{11} - \frac{1}{11} a^{10} - \frac{1}{2} a^{9} - \frac{3}{11} a^{8} + \frac{5}{22} a^{7} + \frac{3}{11} a^{6} - \frac{1}{11} a^{5} - \frac{7}{22} a^{4} - \frac{9}{22} a^{3} + \frac{4}{11} a^{2} + \frac{1}{22} a - \frac{9}{22}$, $\frac{1}{13219148031988162} a^{15} - \frac{190556424315479}{13219148031988162} a^{14} - \frac{78400112273941}{6609574015994081} a^{13} + \frac{1054872853071700}{6609574015994081} a^{12} - \frac{611875655147132}{6609574015994081} a^{11} - \frac{608658747637915}{13219148031988162} a^{10} + \frac{1258159669816475}{13219148031988162} a^{9} + \frac{2056897657150337}{13219148031988162} a^{8} - \frac{4107086826385977}{13219148031988162} a^{7} - \frac{3101190805133001}{6609574015994081} a^{6} + \frac{3686577540168569}{13219148031988162} a^{5} + \frac{1705175267000203}{6609574015994081} a^{4} + \frac{4351568842219359}{13219148031988162} a^{3} - \frac{239399005605299}{1201740730180742} a^{2} - \frac{2668846178491132}{6609574015994081} a - \frac{2673425508202501}{13219148031988162}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 192294.67184 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.32625.1, \(\Q(\zeta_{15})^+\), 4.4.725.1, 8.8.1064390625.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
3Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$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$$[\ ]_{4}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$