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Properties

Label 16.0.11020639142...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 101^{5}$
Root discriminant $56.58$
Ramified primes $2, 5, 101$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![124666421, 0, -96297440, 0, 27412511, 0, -3819820, 0, 337476, 0, -23000, 0, 1146, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 + 1146*x^12 - 23000*x^10 + 337476*x^8 - 3819820*x^6 + 27412511*x^4 - 96297440*x^2 + 124666421)
 
gp: K = bnfinit(x^16 - 40*x^14 + 1146*x^12 - 23000*x^10 + 337476*x^8 - 3819820*x^6 + 27412511*x^4 - 96297440*x^2 + 124666421, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} + 1146 x^{12} - 23000 x^{10} + 337476 x^{8} - 3819820 x^{6} + 27412511 x^{4} - 96297440 x^{2} + 124666421 \)

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:  \(11020639142936576000000000000=2^{32}\cdot 5^{12}\cdot 101^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.58$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{101} a^{12} - \frac{40}{101} a^{10} + \frac{35}{101} a^{8} + \frac{28}{101} a^{6} + \frac{35}{101} a^{4}$, $\frac{1}{1111} a^{13} - \frac{444}{1111} a^{11} + \frac{540}{1111} a^{9} - \frac{73}{1111} a^{7} + \frac{338}{1111} a^{5} - \frac{1}{11} a^{3} + \frac{5}{11} a$, $\frac{1}{100818350734890729412351} a^{14} + \frac{272703596236033184119}{100818350734890729412351} a^{12} + \frac{16184517312529693773273}{100818350734890729412351} a^{10} + \frac{1339043536538894199886}{100818350734890729412351} a^{8} + \frac{7689239599877594349082}{100818350734890729412351} a^{6} - \frac{361964986221830910931}{998201492424660687251} a^{4} + \frac{217322194882077842419}{998201492424660687251} a^{2} + \frac{99573914422348161}{898471190301224741}$, $\frac{1}{100818350734890729412351} a^{15} + \frac{42438688614735236}{9165304612262793582941} a^{13} + \frac{36239292751243331217134}{100818350734890729412351} a^{11} - \frac{44850461885656768510183}{100818350734890729412351} a^{9} + \frac{27562523858150384395261}{100818350734890729412351} a^{7} - \frac{274813280762612111054}{998201492424660687251} a^{5} + \frac{489558965543348938942}{998201492424660687251} a^{3} - \frac{2498571702559069193}{9883183093313472151} a$

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:  \( \frac{238425194664180}{152523979931756020291} a^{14} - \frac{4987996260271767}{152523979931756020291} a^{12} + \frac{147907332614097888}{152523979931756020291} a^{10} - \frac{1891726461978801794}{152523979931756020291} a^{8} + \frac{22078188557103296168}{152523979931756020291} a^{6} - \frac{1453974545779184084}{1510138415165901191} a^{4} - \frac{6347883784777534536}{1510138415165901191} a^{2} + \frac{31752164783954614}{1359260499699281} \) (order $10$)
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:  \( 79651721.7863 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1354:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 59 conjugacy class representatives for t16n1354 are not computed
Character table for t16n1354 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.404000000.2

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 R $16$ R $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.3.4$x^{4} + 808$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$