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Properties

Label 16.0.16244794399...856.14
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 7^{8}$
Root discriminant $18.33$
Ramified primes $2, 3, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 0, -182, 0, 365, 0, -438, 0, 320, 0, -144, 0, 41, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49)
 
gp: K = bnfinit(x^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} + 41 x^{12} - 144 x^{10} + 320 x^{8} - 438 x^{6} + 365 x^{4} - 182 x^{2} + 49 \)

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:  \(162447943996702457856=2^{32}\cdot 3^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.33$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{5502} a^{14} + \frac{419}{5502} a^{12} + \frac{176}{917} a^{10} + \frac{719}{2751} a^{8} - \frac{313}{917} a^{6} - \frac{443}{2751} a^{4} + \frac{1681}{5502} a^{2} - \frac{63}{262}$, $\frac{1}{38514} a^{15} + \frac{1585}{19257} a^{13} - \frac{1658}{6419} a^{11} + \frac{8972}{19257} a^{9} - \frac{2147}{6419} a^{7} - \frac{5945}{19257} a^{5} - \frac{3821}{38514} a^{3} + \frac{34}{917} a$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{83}{786} a^{14} - \frac{77}{131} a^{12} + \frac{1118}{393} a^{10} - \frac{3203}{393} a^{8} + \frac{5248}{393} a^{6} - \frac{5329}{393} a^{4} + \frac{2317}{262} a^{2} - \frac{998}{393} \) (order $6$)
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:  \( 11233.812692 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_4$ (as 16T25):

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 $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), 4.0.1008.2, 4.0.4032.1, 4.0.4032.2, 4.0.1008.1, \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), 8.0.796594176.2, 8.0.260112384.3, 8.0.260112384.7, 8.0.12745506816.12, 8.0.12745506816.18, 8.0.49787136.3, 8.0.796594176.11

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$