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Properties

Label 16.2.13527256579...4751.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,7^{8}\cdot 31^{15}$
Root discriminant $66.18$
Ramified primes $7, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 450, -1002, 35647, -18016, 285015, 90832, -169895, 24690, 2766, 4081, -1565, 120, -15, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 13*x^14 - 15*x^13 + 120*x^12 - 1565*x^11 + 4081*x^10 + 2766*x^9 + 24690*x^8 - 169895*x^7 + 90832*x^6 + 285015*x^5 - 18016*x^4 + 35647*x^3 - 1002*x^2 + 450*x + 81)
 
gp: K = bnfinit(x^16 - 6*x^15 + 13*x^14 - 15*x^13 + 120*x^12 - 1565*x^11 + 4081*x^10 + 2766*x^9 + 24690*x^8 - 169895*x^7 + 90832*x^6 + 285015*x^5 - 18016*x^4 + 35647*x^3 - 1002*x^2 + 450*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 13 x^{14} - 15 x^{13} + 120 x^{12} - 1565 x^{11} + 4081 x^{10} + 2766 x^{9} + 24690 x^{8} - 169895 x^{7} + 90832 x^{6} + 285015 x^{5} - 18016 x^{4} + 35647 x^{3} - 1002 x^{2} + 450 x + 81 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-135272565795848237293667454751=-\,7^{8}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.18$
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:  $7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{21} a^{10} + \frac{1}{21} a^{9} - \frac{1}{21} a^{8} - \frac{1}{7} a^{7} - \frac{1}{21} a^{6} + \frac{5}{21} a^{5} - \frac{2}{21} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{21} a^{11} - \frac{2}{21} a^{9} - \frac{2}{21} a^{8} + \frac{2}{21} a^{7} + \frac{2}{7} a^{6} - \frac{1}{3} a^{5} - \frac{10}{21} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{63} a^{12} + \frac{1}{63} a^{11} - \frac{1}{63} a^{10} - \frac{1}{21} a^{9} - \frac{8}{63} a^{8} - \frac{2}{63} a^{7} + \frac{5}{63} a^{6} - \frac{4}{21} a^{5} - \frac{29}{63} a^{4} - \frac{26}{63} a^{3} - \frac{31}{63} a^{2} - \frac{1}{3} a$, $\frac{1}{63} a^{13} + \frac{1}{63} a^{11} + \frac{1}{63} a^{10} - \frac{8}{63} a^{9} - \frac{1}{21} a^{8} + \frac{4}{63} a^{7} - \frac{2}{63} a^{6} - \frac{23}{63} a^{5} + \frac{10}{21} a^{4} + \frac{13}{63} a^{3} - \frac{26}{63} a^{2} + \frac{4}{21} a$, $\frac{1}{5355} a^{14} + \frac{31}{5355} a^{13} + \frac{1}{5355} a^{12} - \frac{37}{5355} a^{11} - \frac{4}{765} a^{10} - \frac{773}{5355} a^{9} + \frac{877}{5355} a^{8} + \frac{32}{5355} a^{7} - \frac{337}{765} a^{6} - \frac{218}{765} a^{5} + \frac{1021}{5355} a^{4} + \frac{1616}{5355} a^{3} + \frac{667}{5355} a^{2} - \frac{207}{595} a + \frac{71}{595}$, $\frac{1}{172835325743503466878050890835} a^{15} - \frac{202799322071003870404883}{34567065148700693375610178167} a^{14} + \frac{128011190275505329541358931}{34567065148700693375610178167} a^{13} - \frac{1044917073741011660466271493}{172835325743503466878050890835} a^{12} + \frac{3655650490431911042743969184}{172835325743503466878050890835} a^{11} - \frac{726499489509107385418513}{107018777550156945435325629} a^{10} - \frac{17813689120588070393190320}{356361496378357663666084311} a^{9} + \frac{18027242480869345333199659}{290479539064711709038740993} a^{8} - \frac{11168645998974016926489499636}{172835325743503466878050890835} a^{7} + \frac{15070596514384986105816996713}{172835325743503466878050890835} a^{6} - \frac{81559416246843413178405770923}{172835325743503466878050890835} a^{5} + \frac{5197847517459004059431383322}{34567065148700693375610178167} a^{4} + \frac{39113354180085964643878444021}{172835325743503466878050890835} a^{3} - \frac{2565359537909640229479340118}{11522355049566897791870059389} a^{2} + \frac{2026680121177529278566169928}{19203925082611496319783432315} a + \frac{36862470507049167926648383}{304824207660499941583864005}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $8$
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:  \( 1882860909.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{217}) \), 4.2.1459759.1, 8.2.66057786480511.1

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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ $16$ $16$ R $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
7.2.1.2$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}$
31Data not computed