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Properties

Label 16.0.10084035524...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 1361^{3}$
Root discriminant $31.64$
Ramified primes $2, 5, 1361$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group 16T1354

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 15, 30, 70, 76, 72, 140, 45, -140, 72, -76, 70, -30, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 15*x^14 - 30*x^13 + 70*x^12 - 76*x^11 + 72*x^10 - 140*x^9 + 45*x^8 + 140*x^7 + 72*x^6 + 76*x^5 + 70*x^4 + 30*x^3 + 15*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 15*x^14 - 30*x^13 + 70*x^12 - 76*x^11 + 72*x^10 - 140*x^9 + 45*x^8 + 140*x^7 + 72*x^6 + 76*x^5 + 70*x^4 + 30*x^3 + 15*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 15 x^{14} - 30 x^{13} + 70 x^{12} - 76 x^{11} + 72 x^{10} - 140 x^{9} + 45 x^{8} + 140 x^{7} + 72 x^{6} + 76 x^{5} + 70 x^{4} + 30 x^{3} + 15 x^{2} + 2 x + 1 \)

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:  \(1008403552400000000000000=2^{16}\cdot 5^{14}\cdot 1361^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.64$
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:  $2, 5, 1361$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{11} + \frac{1}{20} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{10} a^{5} + \frac{1}{4} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{1}{20}$, $\frac{1}{20} a^{13} + \frac{1}{20} a^{9} - \frac{3}{20} a^{7} - \frac{2}{5} a^{6} + \frac{1}{4} a^{5} - \frac{1}{10} a^{4} - \frac{1}{5} a^{3} + \frac{3}{10} a^{2} + \frac{1}{20} a - \frac{1}{2}$, $\frac{1}{112100} a^{14} + \frac{967}{112100} a^{13} + \frac{251}{28025} a^{12} + \frac{1037}{28025} a^{11} + \frac{1701}{112100} a^{10} + \frac{4471}{112100} a^{9} - \frac{9703}{112100} a^{8} - \frac{31921}{112100} a^{7} - \frac{46347}{112100} a^{6} + \frac{15681}{112100} a^{5} - \frac{26073}{56050} a^{4} - \frac{3531}{56050} a^{3} - \frac{51449}{112100} a^{2} + \frac{12177}{112100} a + \frac{19617}{56050}$, $\frac{1}{112100} a^{15} + \frac{39}{2242} a^{13} - \frac{7}{295} a^{12} + \frac{753}{22420} a^{11} - \frac{934}{28025} a^{10} + \frac{2143}{22420} a^{9} + \frac{87}{5605} a^{8} + \frac{6599}{22420} a^{7} - \frac{2919}{11210} a^{6} - \frac{21469}{56050} a^{5} + \frac{2917}{11210} a^{4} - \frac{3151}{22420} a^{3} - \frac{405}{2242} a^{2} + \frac{2897}{11210} a + \frac{12853}{28025}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( -1 \) (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:  \( 81146.1229534 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.0.5444000000.1

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 $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
1361Data not computed