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Properties

Label 16.4.98340377973...249.30
Degree $16$
Signature $[4, 6]$
Discriminant $13^{10}\cdot 61^{10}$
Root discriminant $64.87$
Ramified primes $13, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27504, -77640, 113084, -112974, 93551, -68932, 43374, -21854, 7780, -1326, -344, 380, -222, 78, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 78*x^13 - 222*x^12 + 380*x^11 - 344*x^10 - 1326*x^9 + 7780*x^8 - 21854*x^7 + 43374*x^6 - 68932*x^5 + 93551*x^4 - 112974*x^3 + 113084*x^2 - 77640*x + 27504)
 
gp: K = bnfinit(x^16 - 4*x^15 - 8*x^14 + 78*x^13 - 222*x^12 + 380*x^11 - 344*x^10 - 1326*x^9 + 7780*x^8 - 21854*x^7 + 43374*x^6 - 68932*x^5 + 93551*x^4 - 112974*x^3 + 113084*x^2 - 77640*x + 27504, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 8 x^{14} + 78 x^{13} - 222 x^{12} + 380 x^{11} - 344 x^{10} - 1326 x^{9} + 7780 x^{8} - 21854 x^{7} + 43374 x^{6} - 68932 x^{5} + 93551 x^{4} - 112974 x^{3} + 113084 x^{2} - 77640 x + 27504 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(98340377973019428085802419249=13^{10}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.87$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{12} a^{10} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{12} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{12} a^{5} + \frac{1}{6} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{312} a^{12} + \frac{5}{156} a^{11} - \frac{11}{312} a^{9} + \frac{3}{26} a^{8} - \frac{1}{39} a^{7} - \frac{29}{312} a^{6} - \frac{23}{156} a^{5} + \frac{5}{78} a^{4} - \frac{113}{312} a^{3} - \frac{17}{39} a^{2} + \frac{1}{26} a + \frac{5}{13}$, $\frac{1}{312} a^{13} + \frac{1}{78} a^{11} - \frac{11}{312} a^{10} - \frac{5}{156} a^{9} - \frac{5}{52} a^{8} - \frac{53}{312} a^{7} - \frac{2}{39} a^{6} + \frac{8}{39} a^{5} - \frac{1}{312} a^{4} - \frac{23}{156} a^{3} + \frac{49}{156} a^{2} + \frac{2}{13}$, $\frac{1}{12168} a^{14} + \frac{1}{4056} a^{13} + \frac{1}{3042} a^{12} + \frac{79}{12168} a^{11} + \frac{295}{12168} a^{10} + \frac{10}{169} a^{9} + \frac{1}{936} a^{8} - \frac{1631}{12168} a^{7} + \frac{329}{3042} a^{6} - \frac{673}{4056} a^{5} - \frac{95}{1352} a^{4} - \frac{629}{1521} a^{3} + \frac{563}{6084} a^{2} + \frac{1273}{3042} a + \frac{71}{507}$, $\frac{1}{6057874781843729832} a^{15} + \frac{8810879928392}{252411449243488743} a^{14} + \frac{269284466901937}{6057874781843729832} a^{13} + \frac{1907783961152879}{3028937390921864916} a^{12} - \frac{60274350620525203}{3028937390921864916} a^{11} - \frac{47506233161053321}{2019291593947909944} a^{10} - \frac{25524884795340754}{757234347730466229} a^{9} + \frac{87231291330121220}{757234347730466229} a^{8} + \frac{296334950331552755}{6057874781843729832} a^{7} + \frac{124830865983880723}{1009645796973954972} a^{6} - \frac{200757872541101153}{1009645796973954972} a^{5} - \frac{547585718607742333}{6057874781843729832} a^{4} - \frac{1007680505829446225}{6057874781843729832} a^{3} + \frac{622248769809144487}{1514468695460932458} a^{2} - \frac{48023199482870983}{504822898486977486} a + \frac{40988746287298066}{84137149747829581}$

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:  $9$
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:  \( 337680929.195 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.10309.1, 8.4.395451064801.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61Data not computed