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Properties

Label 16.0.16244794399...856.15
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 $4$
Class group $[4]$
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, 126, 56, 226, 72, 184, -28, 95, -120, 136, -108, 50, -24, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 24*x^13 + 50*x^12 - 108*x^11 + 136*x^10 - 120*x^9 + 95*x^8 - 28*x^7 + 184*x^6 + 72*x^5 + 226*x^4 + 56*x^3 + 126*x^2 + 49)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 24*x^13 + 50*x^12 - 108*x^11 + 136*x^10 - 120*x^9 + 95*x^8 - 28*x^7 + 184*x^6 + 72*x^5 + 226*x^4 + 56*x^3 + 126*x^2 + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 24 x^{13} + 50 x^{12} - 108 x^{11} + 136 x^{10} - 120 x^{9} + 95 x^{8} - 28 x^{7} + 184 x^{6} + 72 x^{5} + 226 x^{4} + 56 x^{3} + 126 x^{2} + 49 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{22} a^{13} - \frac{1}{22} a^{11} - \frac{2}{11} a^{10} + \frac{1}{22} a^{9} + \frac{2}{11} a^{8} + \frac{1}{22} a^{7} - \frac{2}{11} a^{6} - \frac{1}{22} a^{5} - \frac{4}{11} a^{4} + \frac{1}{22} a^{3} + \frac{4}{11} a^{2} - \frac{4}{11}$, $\frac{1}{5698} a^{14} - \frac{123}{5698} a^{13} - \frac{534}{2849} a^{12} - \frac{166}{2849} a^{11} - \frac{167}{5698} a^{10} + \frac{1201}{5698} a^{9} - \frac{317}{2849} a^{8} + \frac{822}{2849} a^{7} - \frac{675}{5698} a^{6} - \frac{103}{814} a^{5} + \frac{267}{2849} a^{4} - \frac{118}{2849} a^{3} - \frac{195}{2849} a^{2} - \frac{21}{407} a + \frac{20}{407}$, $\frac{1}{591653043674} a^{15} - \frac{10322978}{295826521837} a^{14} - \frac{9834354395}{591653043674} a^{13} - \frac{18236962179}{295826521837} a^{12} + \frac{1710392951}{15990622802} a^{11} + \frac{30931325281}{295826521837} a^{10} - \frac{67596830383}{295826521837} a^{9} - \frac{4922306223}{26893320167} a^{8} + \frac{221805501507}{591653043674} a^{7} + \frac{127506459756}{295826521837} a^{6} - \frac{59805181043}{295826521837} a^{5} - \frac{76420401845}{295826521837} a^{4} - \frac{110271122084}{295826521837} a^{3} - \frac{61038022253}{295826521837} a^{2} + \frac{10506666465}{84521863382} a - \frac{18249590872}{42260931691}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{3574588}{1142187343} a^{15} + \frac{193166350}{7995311401} a^{14} - \frac{579432360}{7995311401} a^{13} + \frac{123352678}{726846491} a^{12} - \frac{276505692}{726846491} a^{11} + \frac{6379228764}{7995311401} a^{10} - \frac{11361779832}{7995311401} a^{9} + \frac{11245280771}{7995311401} a^{8} - \frac{8099908676}{7995311401} a^{7} + \frac{5653367884}{7995311401} a^{6} - \frac{916537608}{1142187343} a^{5} + \frac{1263822514}{726846491} a^{4} + \frac{4936397744}{7995311401} a^{3} + \frac{20607684606}{7995311401} a^{2} + \frac{122413616}{1142187343} a + \frac{1679138947}{1142187343} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4446.01541964 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{7}) \), 4.2.4032.1, 4.2.1792.1, 4.2.448.1, 4.2.16128.1, \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{7})\), 8.0.12745506816.4, 8.0.12745506816.16, 8.0.12745506816.19, 8.0.16257024.1, 8.0.260112384.4, 8.4.157351936.1, 8.4.12745506816.3

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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$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}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$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}$