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Properties

Label 16.0.11329274025...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 29^{4}$
Root discriminant $13.44$
Ramified primes $3, 5, 29$
Class number $1$
Class group Trivial
Galois group $C_2 \times (C_2^2:C_4)$ (as 16T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 4, 9, 12, -12, 8, 54, -19, -54, 8, 12, 12, -9, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 + 4*x^14 - 9*x^13 + 12*x^12 + 12*x^11 + 8*x^10 - 54*x^9 - 19*x^8 + 54*x^7 + 8*x^6 - 12*x^5 + 12*x^4 + 9*x^3 + 4*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 4 x^{14} - 9 x^{13} + 12 x^{12} + 12 x^{11} + 8 x^{10} - 54 x^{9} - 19 x^{8} + 54 x^{7} + 8 x^{6} - 12 x^{5} + 12 x^{4} + 9 x^{3} + 4 x^{2} + 3 x + 1 \)

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:  \(1132927402587890625=3^{8}\cdot 5^{12}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.44$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{20} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{10} a^{9} + \frac{1}{4} a^{7} - \frac{1}{20} a^{6} - \frac{1}{4} a^{5} + \frac{1}{10} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{20}$, $\frac{1}{20} a^{13} + \frac{3}{20} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{5} a^{7} + \frac{1}{4} a^{5} + \frac{1}{10} a^{4} + \frac{1}{4} a^{3} - \frac{1}{5} a - \frac{1}{4}$, $\frac{1}{2360} a^{14} - \frac{13}{1180} a^{13} + \frac{13}{2360} a^{12} + \frac{69}{1180} a^{11} + \frac{37}{2360} a^{10} + \frac{479}{2360} a^{9} + \frac{237}{1180} a^{8} - \frac{919}{2360} a^{7} + \frac{44}{295} a^{6} - \frac{583}{2360} a^{5} + \frac{553}{2360} a^{4} - \frac{231}{590} a^{3} + \frac{931}{2360} a^{2} + \frac{141}{590} a + \frac{353}{2360}$, $\frac{1}{342200} a^{15} + \frac{7}{171100} a^{14} - \frac{6573}{342200} a^{13} - \frac{15}{1711} a^{12} - \frac{54033}{342200} a^{11} - \frac{62469}{342200} a^{10} + \frac{10601}{34220} a^{9} + \frac{77631}{342200} a^{8} + \frac{71299}{171100} a^{7} - \frac{11201}{68440} a^{6} + \frac{156593}{342200} a^{5} + \frac{5983}{42775} a^{4} - \frac{483}{13688} a^{3} + \frac{51647}{171100} a^{2} + \frac{116487}{342200} a + \frac{4479}{42775}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{18133}{342200} a^{15} + \frac{6001}{171100} a^{14} - \frac{162059}{342200} a^{13} + \frac{20307}{34220} a^{12} - \frac{501719}{342200} a^{11} + \frac{1283023}{342200} a^{10} + \frac{63899}{34220} a^{9} - \frac{1073567}{342200} a^{8} - \frac{1003129}{85550} a^{7} + \frac{259233}{68440} a^{6} + \frac{4734049}{342200} a^{5} - \frac{4104}{725} a^{4} - \frac{70737}{68440} a^{3} + \frac{159699}{42775} a^{2} - \frac{30179}{342200} a + \frac{19319}{85550} \) (order $30$)
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:  \( 3158.73230855 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_4$ (as 16T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_2^2:C_4)$
Character table for $C_2 \times (C_2^2:C_4)$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{5})\), 4.4.32625.1, \(\Q(\zeta_{15})^+\), 4.0.3625.1, 4.0.6525.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.4.725.1, 8.0.42575625.1, 8.0.1064390625.3, 8.0.1064390625.1, \(\Q(\zeta_{15})\), 8.0.1064390625.2, 8.0.13140625.1, 8.8.1064390625.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$