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Properties

Label 16.0.35256881742...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{8}\cdot 19^{8}$
Root discriminant $22.22$
Ramified primes $3, 5, 19$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -427, 1944, -6083, 13680, -22494, 27823, -26498, 19974, -12229, 6166, -2544, 879, -238, 54, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 54*x^14 - 238*x^13 + 879*x^12 - 2544*x^11 + 6166*x^10 - 12229*x^9 + 19974*x^8 - 26498*x^7 + 27823*x^6 - 22494*x^5 + 13680*x^4 - 6083*x^3 + 1944*x^2 - 427*x + 49)
 
gp: K = bnfinit(x^16 - 8*x^15 + 54*x^14 - 238*x^13 + 879*x^12 - 2544*x^11 + 6166*x^10 - 12229*x^9 + 19974*x^8 - 26498*x^7 + 27823*x^6 - 22494*x^5 + 13680*x^4 - 6083*x^3 + 1944*x^2 - 427*x + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 54 x^{14} - 238 x^{13} + 879 x^{12} - 2544 x^{11} + 6166 x^{10} - 12229 x^{9} + 19974 x^{8} - 26498 x^{7} + 27823 x^{6} - 22494 x^{5} + 13680 x^{4} - 6083 x^{3} + 1944 x^{2} - 427 x + 49 \)

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:  \(3525688174246906640625=3^{12}\cdot 5^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.22$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1555988} a^{14} - \frac{1}{222284} a^{13} + \frac{11969}{777994} a^{12} - \frac{143537}{1555988} a^{11} + \frac{323507}{1555988} a^{10} - \frac{150973}{777994} a^{9} - \frac{760433}{1555988} a^{8} + \frac{162811}{1555988} a^{7} + \frac{196935}{777994} a^{6} + \frac{114973}{1555988} a^{5} - \frac{135461}{1555988} a^{4} - \frac{31059}{111142} a^{3} + \frac{513137}{1555988} a^{2} + \frac{243973}{1555988} a + \frac{53609}{111142}$, $\frac{1}{45123652} a^{15} + \frac{1}{6446236} a^{14} - \frac{4255127}{45123652} a^{13} - \frac{98701}{22561826} a^{12} - \frac{21913855}{45123652} a^{11} + \frac{6950131}{45123652} a^{10} + \frac{5074282}{11280913} a^{9} - \frac{18263191}{45123652} a^{8} - \frac{5495713}{45123652} a^{7} - \frac{2436883}{22561826} a^{6} + \frac{15478053}{45123652} a^{5} + \frac{2389939}{6446236} a^{4} + \frac{3631237}{22561826} a^{3} - \frac{12799953}{45123652} a^{2} - \frac{16117}{6446236} a - \frac{777359}{6446236}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( \frac{47797}{52838} a^{15} - \frac{716955}{105676} a^{14} + \frac{4803149}{105676} a^{13} - \frac{20346651}{105676} a^{12} + \frac{73832993}{105676} a^{11} - \frac{206190699}{105676} a^{10} + \frac{486040425}{105676} a^{9} - \frac{925209129}{105676} a^{8} + \frac{1444999111}{105676} a^{7} - \frac{1807330053}{105676} a^{6} + \frac{1751293779}{105676} a^{5} - \frac{1269127775}{105676} a^{4} + \frac{668242023}{105676} a^{3} - \frac{244160857}{105676} a^{2} + \frac{62178375}{105676} a - \frac{9148827}{105676} \) (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:  \( 11398.5231056 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_2$ (as 16T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{285}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{-95})\), \(\Q(\sqrt{5}, \sqrt{57})\), \(\Q(\sqrt{-15}, \sqrt{-19})\), \(\Q(\sqrt{-3}, \sqrt{-19})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-15}, \sqrt{57})\), \(\Q(\sqrt{5}, \sqrt{-19})\), 8.0.6597500625.1, 8.4.59377505625.1 x2, 8.0.2375100225.1 x2, 8.0.164480625.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$