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Properties

Label 16.8.92209158553...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 5^{12}\cdot 11^{4}\cdot 31^{2}$
Root discriminant $31.46$
Ramified primes $2, 5, 11, 31$
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![25, -150, 390, -890, 1626, -1208, -424, 814, -229, 196, -144, -18, 4, 10, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 10*x^13 + 4*x^12 - 18*x^11 - 144*x^10 + 196*x^9 - 229*x^8 + 814*x^7 - 424*x^6 - 1208*x^5 + 1626*x^4 - 890*x^3 + 390*x^2 - 150*x + 25)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 10*x^13 + 4*x^12 - 18*x^11 - 144*x^10 + 196*x^9 - 229*x^8 + 814*x^7 - 424*x^6 - 1208*x^5 + 1626*x^4 - 890*x^3 + 390*x^2 - 150*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 10 x^{13} + 4 x^{12} - 18 x^{11} - 144 x^{10} + 196 x^{9} - 229 x^{8} + 814 x^{7} - 424 x^{6} - 1208 x^{5} + 1626 x^{4} - 890 x^{3} + 390 x^{2} - 150 x + 25 \)

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:  \(922091585536000000000000=2^{28}\cdot 5^{12}\cdot 11^{4}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.46$
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:  $2, 5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{45} a^{12} - \frac{1}{9} a^{11} - \frac{7}{45} a^{10} + \frac{7}{45} a^{9} - \frac{2}{9} a^{8} + \frac{1}{5} a^{7} - \frac{8}{45} a^{6} - \frac{2}{45} a^{5} - \frac{1}{3} a^{4} + \frac{4}{15} a^{3} + \frac{14}{45} a^{2} + \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{45} a^{13} - \frac{2}{45} a^{11} + \frac{2}{45} a^{10} - \frac{4}{9} a^{9} + \frac{4}{45} a^{8} + \frac{22}{45} a^{7} + \frac{2}{5} a^{6} + \frac{1}{9} a^{5} - \frac{2}{5} a^{4} - \frac{16}{45} a^{3} - \frac{2}{9} a^{2} + \frac{4}{9}$, $\frac{1}{135} a^{14} - \frac{8}{135} a^{11} - \frac{19}{135} a^{10} + \frac{16}{45} a^{9} - \frac{13}{135} a^{8} + \frac{22}{45} a^{7} + \frac{34}{135} a^{6} - \frac{37}{135} a^{5} + \frac{44}{135} a^{4} + \frac{14}{135} a^{3} + \frac{28}{135} a^{2} - \frac{1}{27} a - \frac{11}{27}$, $\frac{1}{125792609075235} a^{15} - \frac{31798161109}{125792609075235} a^{14} - \frac{99938259034}{41930869691745} a^{13} + \frac{909273019528}{125792609075235} a^{12} + \frac{8088507357727}{125792609075235} a^{11} + \frac{9873974196043}{125792609075235} a^{10} - \frac{44541838210663}{125792609075235} a^{9} - \frac{3796578897580}{25158521815047} a^{8} + \frac{5550647071421}{25158521815047} a^{7} + \frac{30325700067913}{125792609075235} a^{6} - \frac{3486314148298}{8386173938349} a^{5} - \frac{8446432659457}{41930869691745} a^{4} - \frac{34437930168844}{125792609075235} a^{3} + \frac{16799923096394}{41930869691745} a^{2} - \frac{3020607655804}{25158521815047} a + \frac{1750481668295}{25158521815047}$

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:  \( 1739973.92982 \) (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

\(\Q(\sqrt{5}) \), 4.4.4400.1, \(\Q(\zeta_{20})^+\), 4.4.22000.1, 8.8.7744000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.2.0.1$x^{2} - x + 12$$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$$[\ ]^{2}$
31.4.2.2$x^{4} - 31 x^{2} + 11532$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$