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Properties

Label 16.4.12780765970...3632.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{67}\cdot 17^{6}\cdot 599^{2}$
Root discriminant $117.26$
Ramified primes $2, 17, 599$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1113

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![829547912, 0, -446748576, 0, -346080720, 0, -57999200, 0, -3717076, 0, -70560, 0, 2020, 0, 96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 96*x^14 + 2020*x^12 - 70560*x^10 - 3717076*x^8 - 57999200*x^6 - 346080720*x^4 - 446748576*x^2 + 829547912)
 
gp: K = bnfinit(x^16 + 96*x^14 + 2020*x^12 - 70560*x^10 - 3717076*x^8 - 57999200*x^6 - 346080720*x^4 - 446748576*x^2 + 829547912, 1)
 

Normalized defining polynomial

\( x^{16} + 96 x^{14} + 2020 x^{12} - 70560 x^{10} - 3717076 x^{8} - 57999200 x^{6} - 346080720 x^{4} - 446748576 x^{2} + 829547912 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1278076597085555502017391343173632=2^{67}\cdot 17^{6}\cdot 599^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.26$
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:  $2, 17, 599$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{34} a^{10} + \frac{5}{34} a^{8} - \frac{4}{17} a^{6} + \frac{5}{17} a^{4} - \frac{7}{17} a^{2}$, $\frac{1}{68} a^{11} - \frac{3}{17} a^{9} - \frac{2}{17} a^{7} - \frac{6}{17} a^{5} + \frac{5}{17} a^{3}$, $\frac{1}{68} a^{12} - \frac{4}{17} a^{8} + \frac{4}{17} a^{6} + \frac{1}{17} a^{4} - \frac{8}{17} a^{2}$, $\frac{1}{68} a^{13} - \frac{4}{17} a^{9} + \frac{4}{17} a^{7} + \frac{1}{17} a^{5} - \frac{8}{17} a^{3}$, $\frac{1}{16096434166617251908132699396} a^{14} - \frac{14202918200217842238156162}{4024108541654312977033174849} a^{12} - \frac{105022778406238953857260231}{8048217083308625954066349698} a^{10} + \frac{1111851328351910721756882465}{8048217083308625954066349698} a^{8} + \frac{4067786384239483045454799}{4024108541654312977033174849} a^{6} + \frac{1637609571712668812823394356}{4024108541654312977033174849} a^{4} - \frac{782501686178909252629055419}{4024108541654312977033174849} a^{2} - \frac{98123733573169000330338}{395179077055318960722103}$, $\frac{1}{16096434166617251908132699396} a^{15} - \frac{14202918200217842238156162}{4024108541654312977033174849} a^{13} + \frac{26666710343658149758019235}{16096434166617251908132699396} a^{11} - \frac{308422274584905623078355717}{8048217083308625954066349698} a^{9} - \frac{469356747928032631899624595}{4024108541654312977033174849} a^{7} + \frac{217335968775852467988156174}{4024108541654312977033174849} a^{5} + \frac{401059649601771034733643066}{4024108541654312977033174849} a^{3} - \frac{98123733573169000330338}{395179077055318960722103} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 11780892895.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1113:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1113
Character table for t16n1113 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 8.8.9697230848.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
599Data not computed