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Properties

Label 16.0.76060599125...6849.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 23^{14}$
Root discriminant $26.92$
Ramified primes $3, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1296, -324, -2115, 1899, 1543, -1930, -1206, 682, 1120, -384, -224, -30, 78, 2, -5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 5*x^14 + 2*x^13 + 78*x^12 - 30*x^11 - 224*x^10 - 384*x^9 + 1120*x^8 + 682*x^7 - 1206*x^6 - 1930*x^5 + 1543*x^4 + 1899*x^3 - 2115*x^2 - 324*x + 1296)
 
gp: K = bnfinit(x^16 - 3*x^15 - 5*x^14 + 2*x^13 + 78*x^12 - 30*x^11 - 224*x^10 - 384*x^9 + 1120*x^8 + 682*x^7 - 1206*x^6 - 1930*x^5 + 1543*x^4 + 1899*x^3 - 2115*x^2 - 324*x + 1296, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 5 x^{14} + 2 x^{13} + 78 x^{12} - 30 x^{11} - 224 x^{10} - 384 x^{9} + 1120 x^{8} + 682 x^{7} - 1206 x^{6} - 1930 x^{5} + 1543 x^{4} + 1899 x^{3} - 2115 x^{2} - 324 x + 1296 \)

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:  \(76060599125298737496849=3^{8}\cdot 23^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.92$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{7} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{12} a$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{9} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{5}{12} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{10} + \frac{1}{12} a^{4} + \frac{1}{6} a$, $\frac{1}{48} a^{11} + \frac{1}{48} a^{8} + \frac{1}{6} a^{6} + \frac{11}{48} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{7}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{240} a^{12} + \frac{1}{240} a^{9} + \frac{7}{48} a^{6} - \frac{1}{6} a^{4} + \frac{67}{240} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{2}{5}$, $\frac{1}{2640} a^{13} - \frac{1}{660} a^{12} - \frac{1}{132} a^{11} - \frac{59}{2640} a^{10} + \frac{7}{330} a^{9} - \frac{1}{44} a^{8} - \frac{7}{176} a^{7} + \frac{1}{12} a^{6} + \frac{19}{132} a^{5} - \frac{51}{880} a^{4} - \frac{53}{165} a^{3} - \frac{17}{44} a^{2} - \frac{17}{220} a - \frac{13}{55}$, $\frac{1}{15840} a^{14} - \frac{7}{7920} a^{12} + \frac{13}{7920} a^{11} - \frac{1}{88} a^{10} + \frac{31}{2640} a^{9} - \frac{31}{792} a^{8} - \frac{7}{264} a^{7} + \frac{247}{1584} a^{6} + \frac{1151}{7920} a^{5} - \frac{17}{264} a^{4} - \frac{809}{7920} a^{3} + \frac{2371}{15840} a^{2} - \frac{13}{264} a - \frac{21}{110}$, $\frac{1}{73038240} a^{15} + \frac{13}{1623072} a^{14} - \frac{185}{7303824} a^{13} - \frac{3734}{2282445} a^{12} - \frac{2137}{304326} a^{11} - \frac{92689}{2434608} a^{10} + \frac{635693}{36519120} a^{9} + \frac{94451}{2434608} a^{8} + \frac{103855}{7303824} a^{7} + \frac{283466}{2282445} a^{6} + \frac{62687}{304326} a^{5} + \frac{1429367}{7303824} a^{4} - \frac{21415787}{73038240} a^{3} - \frac{1132237}{4869216} a^{2} - \frac{16547}{135256} a + \frac{9073}{169070}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{2801}{2434608} a^{15} + \frac{12403}{24346080} a^{14} + \frac{129149}{12173040} a^{13} + \frac{10713}{676280} a^{12} - \frac{812291}{12173040} a^{11} - \frac{53957}{368880} a^{10} + \frac{202093}{1521630} a^{9} + \frac{231707}{304326} a^{8} + \frac{327289}{2434608} a^{7} - \frac{2390081}{1217304} a^{6} - \frac{9618577}{12173040} a^{5} + \frac{23142793}{12173040} a^{4} + \frac{7817551}{4057680} a^{3} - \frac{68422667}{24346080} a^{2} - \frac{276027}{676280} a + \frac{456557}{169070} \) (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:  \( 561593.849789 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-3}, \sqrt{-23})\), 4.2.109503.1 x2, 4.0.36501.1 x2, 8.0.11990907009.1, 8.2.275790861207.1 x4, 8.0.91930287069.1 x4

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
23Data not computed