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Properties

Label 16.16.3272967098...0896.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 11^{8}\cdot 13^{12}$
Root discriminant $45.41$
Ramified primes $2, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_4:C_4$ (as 16T62)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -36, -126, 405, 1160, -1684, -3560, 2657, 3560, -1684, -1160, 405, 126, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 36*x^14 + 126*x^13 + 405*x^12 - 1160*x^11 - 1684*x^10 + 3560*x^9 + 2657*x^8 - 3560*x^7 - 1684*x^6 + 1160*x^5 + 405*x^4 - 126*x^3 - 36*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 36*x^14 + 126*x^13 + 405*x^12 - 1160*x^11 - 1684*x^10 + 3560*x^9 + 2657*x^8 - 3560*x^7 - 1684*x^6 + 1160*x^5 + 405*x^4 - 126*x^3 - 36*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 36 x^{14} + 126 x^{13} + 405 x^{12} - 1160 x^{11} - 1684 x^{10} + 3560 x^{9} + 2657 x^{8} - 3560 x^{7} - 1684 x^{6} + 1160 x^{5} + 405 x^{4} - 126 x^{3} - 36 x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(327296709839930989200080896=2^{16}\cdot 11^{8}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.41$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{9} + \frac{1}{10} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{10}$, $\frac{1}{10} a^{11} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{10} a$, $\frac{1}{40} a^{12} + \frac{1}{40} a^{11} - \frac{1}{20} a^{10} + \frac{1}{40} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} + \frac{9}{20} a^{5} - \frac{7}{40} a^{4} + \frac{1}{5} a^{3} + \frac{7}{20} a^{2} - \frac{13}{40} a + \frac{1}{40}$, $\frac{1}{40} a^{13} + \frac{1}{40} a^{11} - \frac{1}{20} a^{10} + \frac{1}{40} a^{9} + \frac{1}{40} a^{8} + \frac{19}{40} a^{7} - \frac{11}{40} a^{6} - \frac{1}{40} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{7}{40} a^{2} - \frac{3}{20} a + \frac{3}{8}$, $\frac{1}{9480} a^{14} + \frac{23}{1896} a^{13} - \frac{4}{395} a^{12} - \frac{11}{2370} a^{11} - \frac{187}{9480} a^{10} - \frac{39}{790} a^{9} + \frac{53}{1896} a^{8} - \frac{382}{1185} a^{7} - \frac{1687}{9480} a^{6} + \frac{633}{1580} a^{5} + \frac{469}{948} a^{4} + \frac{667}{9480} a^{3} - \frac{679}{3160} a^{2} - \frac{1009}{4740} a + \frac{77}{237}$, $\frac{1}{112897320} a^{15} - \frac{811}{22579464} a^{14} + \frac{148297}{37632440} a^{13} + \frac{153857}{22579464} a^{12} + \frac{1272071}{112897320} a^{11} - \frac{771029}{18816220} a^{10} + \frac{1075571}{56448660} a^{9} + \frac{367969}{28224330} a^{8} - \frac{19914161}{56448660} a^{7} - \frac{988727}{3763244} a^{6} - \frac{1838209}{22579464} a^{5} + \frac{27684103}{112897320} a^{4} - \frac{15520531}{37632440} a^{3} - \frac{32814221}{112897320} a^{2} + \frac{34472261}{112897320} a - \frac{1382109}{4704055}$

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:  $15$
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:  \( 153260041.732 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4:C_4$ (as 16T62):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $A_4:C_4$
Character table for $A_4:C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.4253392.1, 4.4.35152.1, 8.8.18091343505664.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 24 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.12.8.1$x^{12} - 33 x^{9} + 363 x^{6} - 1331 x^{3} + 117128$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$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}$