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Properties

Label 16.8.39228809359...4849.2
Degree $16$
Signature $[8, 4]$
Discriminant $13^{12}\cdot 17^{14}$
Root discriminant $81.68$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3247, -35088, 36567, 44999, -268923, 37621, 104159, -8721, 7175, -10079, 1247, -31, -457, 133, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 11*x^14 + 133*x^13 - 457*x^12 - 31*x^11 + 1247*x^10 - 10079*x^9 + 7175*x^8 - 8721*x^7 + 104159*x^6 + 37621*x^5 - 268923*x^4 + 44999*x^3 + 36567*x^2 - 35088*x - 3247)
 
gp: K = bnfinit(x^16 - 2*x^15 - 11*x^14 + 133*x^13 - 457*x^12 - 31*x^11 + 1247*x^10 - 10079*x^9 + 7175*x^8 - 8721*x^7 + 104159*x^6 + 37621*x^5 - 268923*x^4 + 44999*x^3 + 36567*x^2 - 35088*x - 3247, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 11 x^{14} + 133 x^{13} - 457 x^{12} - 31 x^{11} + 1247 x^{10} - 10079 x^{9} + 7175 x^{8} - 8721 x^{7} + 104159 x^{6} + 37621 x^{5} - 268923 x^{4} + 44999 x^{3} + 36567 x^{2} - 35088 x - 3247 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3922880935919264967742950184849=13^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.68$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{503933661311866601038529451230931122252211074} a^{15} - \frac{29609469922766437827648557264185312977431489}{503933661311866601038529451230931122252211074} a^{14} + \frac{8750220043024164511410516313378955786191595}{503933661311866601038529451230931122252211074} a^{13} + \frac{22622590860498120289670720987879634990096691}{503933661311866601038529451230931122252211074} a^{12} - \frac{26499817644480606509309494543043596930546563}{503933661311866601038529451230931122252211074} a^{11} - \frac{229861441842696898021970475449462381116680947}{503933661311866601038529451230931122252211074} a^{10} + \frac{204059666122815138534458949105872661472376989}{503933661311866601038529451230931122252211074} a^{9} - \frac{198089231835243981632818024485191411719272049}{503933661311866601038529451230931122252211074} a^{8} + \frac{197333147390827091508576297705834380809853001}{503933661311866601038529451230931122252211074} a^{7} + \frac{20542303304633837721174018812825800799070037}{503933661311866601038529451230931122252211074} a^{6} - \frac{152519992531281256095733905243262596607975857}{503933661311866601038529451230931122252211074} a^{5} + \frac{358014505221057188488696010936985245950651}{1351028582605540485357987804908662526145338} a^{4} - \frac{124167873600669889538039501552468165615550367}{503933661311866601038529451230931122252211074} a^{3} + \frac{68454006097595200026057265430232762516698289}{503933661311866601038529451230931122252211074} a^{2} - \frac{235205177714097957990639327178238326669639223}{503933661311866601038529451230931122252211074} a + \frac{123309214610703900534257141719383806800884536}{251966830655933300519264725615465561126105537}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $11$
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:  \( 636412697.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

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 $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{221}) \), 4.4.830297.1, 4.4.4913.1, \(\Q(\sqrt{13}, \sqrt{17})\), 8.8.689393108209.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$