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Properties

Label 16.0.58264304400...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 149^{2}$
Root discriminant $26.48$
Ramified primes $2, 3, 5, 149$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1720

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61, -426, 1279, -2066, 2096, -1938, 1896, -1098, 1022, -552, 426, -202, 131, -44, 19, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 19*x^14 - 44*x^13 + 131*x^12 - 202*x^11 + 426*x^10 - 552*x^9 + 1022*x^8 - 1098*x^7 + 1896*x^6 - 1938*x^5 + 2096*x^4 - 2066*x^3 + 1279*x^2 - 426*x + 61)
 
gp: K = bnfinit(x^16 - 4*x^15 + 19*x^14 - 44*x^13 + 131*x^12 - 202*x^11 + 426*x^10 - 552*x^9 + 1022*x^8 - 1098*x^7 + 1896*x^6 - 1938*x^5 + 2096*x^4 - 2066*x^3 + 1279*x^2 - 426*x + 61, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 19 x^{14} - 44 x^{13} + 131 x^{12} - 202 x^{11} + 426 x^{10} - 552 x^{9} + 1022 x^{8} - 1098 x^{7} + 1896 x^{6} - 1938 x^{5} + 2096 x^{4} - 2066 x^{3} + 1279 x^{2} - 426 x + 61 \)

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:  \(58264304400000000000000=2^{16}\cdot 3^{8}\cdot 5^{14}\cdot 149^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.48$
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:  $2, 3, 5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} - \frac{9}{31} a^{13} - \frac{12}{31} a^{12} - \frac{13}{31} a^{11} - \frac{8}{31} a^{10} - \frac{11}{31} a^{9} + \frac{4}{31} a^{8} - \frac{15}{31} a^{7} - \frac{13}{31} a^{6} + \frac{14}{31} a^{5} - \frac{7}{31} a^{4} + \frac{9}{31} a^{3} + \frac{10}{31} a^{2} - \frac{10}{31} a + \frac{11}{31}$, $\frac{1}{114914387664612541321} a^{15} + \frac{570937318112677029}{114914387664612541321} a^{14} - \frac{35852803610959241111}{114914387664612541321} a^{13} + \frac{19523630542845414293}{114914387664612541321} a^{12} + \frac{2457429348080477022}{114914387664612541321} a^{11} - \frac{1111935577640081709}{3706915731116533591} a^{10} - \frac{30966652422401667850}{114914387664612541321} a^{9} + \frac{19486439507497302897}{114914387664612541321} a^{8} + \frac{50266737969062227328}{114914387664612541321} a^{7} - \frac{51214985919970543679}{114914387664612541321} a^{6} - \frac{36823921101690825027}{114914387664612541321} a^{5} - \frac{41182753644517666172}{114914387664612541321} a^{4} + \frac{26459763068005810376}{114914387664612541321} a^{3} - \frac{30826457747582481432}{114914387664612541321} a^{2} + \frac{28047812995933872008}{114914387664612541321} a - \frac{8338163040122698743}{114914387664612541321}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 22674.3470748 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1720:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 116 conjugacy class representatives for t16n1720 are not computed
Character table for t16n1720 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.1620000000.1

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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
5Data not computed
149Data not computed