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Properties

Label 16.0.13083427795...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 11^{8}$
Root discriminant $13.56$
Ramified primes $5, 11$
Class number $2$
Class group $[2]$
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 8, -10, 28, -25, 36, -50, 35, -50, 36, -25, 28, -10, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 - 10*x^13 + 28*x^12 - 25*x^11 + 36*x^10 - 50*x^9 + 35*x^8 - 50*x^7 + 36*x^6 - 25*x^5 + 28*x^4 - 10*x^3 + 8*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} - 10 x^{13} + 28 x^{12} - 25 x^{11} + 36 x^{10} - 50 x^{9} + 35 x^{8} - 50 x^{7} + 36 x^{6} - 25 x^{5} + 28 x^{4} - 10 x^{3} + 8 x^{2} - 5 x + 1 \)

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:  \(1308342779541015625=5^{14}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.56$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{239} a^{14} + \frac{109}{239} a^{13} + \frac{5}{239} a^{12} - \frac{27}{239} a^{11} + \frac{52}{239} a^{10} - \frac{45}{239} a^{9} + \frac{112}{239} a^{8} + \frac{96}{239} a^{7} + \frac{112}{239} a^{6} - \frac{45}{239} a^{5} + \frac{52}{239} a^{4} - \frac{27}{239} a^{3} + \frac{5}{239} a^{2} + \frac{109}{239} a + \frac{1}{239}$, $\frac{1}{4541} a^{15} - \frac{6}{4541} a^{14} - \frac{1297}{4541} a^{13} - \frac{841}{4541} a^{12} - \frac{1145}{4541} a^{11} + \frac{189}{4541} a^{10} - \frac{210}{4541} a^{9} + \frac{122}{4541} a^{8} - \frac{1607}{4541} a^{7} - \frac{1692}{4541} a^{6} + \frac{208}{4541} a^{5} - \frac{271}{4541} a^{4} + \frac{242}{4541} a^{3} - \frac{1183}{4541} a^{2} - \frac{823}{4541} a - \frac{1310}{4541}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{791}{4541} a^{15} - \frac{2409}{4541} a^{14} + \frac{776}{4541} a^{13} - \frac{8724}{4541} a^{12} + \frac{34767}{4541} a^{11} - \frac{10519}{4541} a^{10} + \frac{55677}{4541} a^{9} - \frac{73149}{4541} a^{8} + \frac{43055}{4541} a^{7} - \frac{109395}{4541} a^{6} + \frac{54822}{4541} a^{5} - \frac{61050}{4541} a^{4} + \frac{73831}{4541} a^{3} - \frac{11327}{4541} a^{2} + \frac{30594}{4541} a - \frac{12148}{4541} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 826.91591824 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{5})\), 4.4.15125.1, \(\Q(\sqrt{5}, \sqrt{-11})\), 8.0.9453125.1 x2, 8.4.1143828125.1 x2, 8.0.228765625.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$