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Properties

Label 16.0.31037260546...2976.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{58}\cdot 3^{4}\cdot 1153^{2}$
Root discriminant $39.20$
Ramified primes $2, 3, 1153$
Class number $16$ (GRH)
Class group $[16]$ (GRH)
Galois group 16T1385

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![353, 96, -984, -160, 1372, -256, -776, 864, 714, -352, 680, -64, 212, 0, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 + 212*x^12 - 64*x^11 + 680*x^10 - 352*x^9 + 714*x^8 + 864*x^7 - 776*x^6 - 256*x^5 + 1372*x^4 - 160*x^3 - 984*x^2 + 96*x + 353)
 
gp: K = bnfinit(x^16 + 24*x^14 + 212*x^12 - 64*x^11 + 680*x^10 - 352*x^9 + 714*x^8 + 864*x^7 - 776*x^6 - 256*x^5 + 1372*x^4 - 160*x^3 - 984*x^2 + 96*x + 353, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} + 212 x^{12} - 64 x^{11} + 680 x^{10} - 352 x^{9} + 714 x^{8} + 864 x^{7} - 776 x^{6} - 256 x^{5} + 1372 x^{4} - 160 x^{3} - 984 x^{2} + 96 x + 353 \)

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:  \(31037260546487147588222976=2^{58}\cdot 3^{4}\cdot 1153^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.20$
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:  $2, 3, 1153$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{12} a^{12} + \frac{1}{12} a^{10} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{12} a^{4} - \frac{1}{3} a^{3} + \frac{5}{12} a^{2} + \frac{1}{6}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{5}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{12} a^{3} - \frac{1}{2} a^{2} + \frac{5}{12} a - \frac{1}{4}$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{5}{12} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{5}{12}$, $\frac{1}{119100152877846124044} a^{15} - \frac{4883821172251547017}{119100152877846124044} a^{14} - \frac{2662192381413426313}{119100152877846124044} a^{13} - \frac{138540841075019016}{9925012739820510337} a^{12} - \frac{2238535366642292967}{39700050959282041348} a^{11} + \frac{14211214401422245051}{119100152877846124044} a^{10} - \frac{4702484267791327005}{39700050959282041348} a^{9} - \frac{7439688695008800317}{59550076438923062022} a^{8} + \frac{47119292199776478653}{119100152877846124044} a^{7} - \frac{21903940983391422181}{119100152877846124044} a^{6} - \frac{22267774180695486841}{119100152877846124044} a^{5} - \frac{12109742382608305277}{29775038219461531011} a^{4} + \frac{1507037137888249777}{39700050959282041348} a^{3} + \frac{12789487307630173805}{39700050959282041348} a^{2} + \frac{28915251805479450689}{119100152877846124044} a + \frac{1505427321671502031}{59550076438923062022}$

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

Class group and class number

$C_{16}$, which has order $16$ (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:  $7$
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:  \( 108044.44759 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1385:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1385 are not computed
Character table for t16n1385 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.77376520192.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
1153Data not computed