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Properties

Label 16.4.25551760733...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 5^{11}\cdot 41^{8}$
Root discriminant $38.72$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1188

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![695, -5, -1870, -1620, 734, 1839, 524, 561, -625, -339, -120, -29, 40, 6, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 + 6*x^13 + 40*x^12 - 29*x^11 - 120*x^10 - 339*x^9 - 625*x^8 + 561*x^7 + 524*x^6 + 1839*x^5 + 734*x^4 - 1620*x^3 - 1870*x^2 - 5*x + 695)
 
gp: K = bnfinit(x^16 - 3*x^15 + 6*x^14 + 6*x^13 + 40*x^12 - 29*x^11 - 120*x^10 - 339*x^9 - 625*x^8 + 561*x^7 + 524*x^6 + 1839*x^5 + 734*x^4 - 1620*x^3 - 1870*x^2 - 5*x + 695, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 6 x^{14} + 6 x^{13} + 40 x^{12} - 29 x^{11} - 120 x^{10} - 339 x^{9} - 625 x^{8} + 561 x^{7} + 524 x^{6} + 1839 x^{5} + 734 x^{4} - 1620 x^{3} - 1870 x^{2} - 5 x + 695 \)

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:  \(25551760733187200000000000=2^{16}\cdot 5^{11}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.72$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{60} a^{14} + \frac{1}{6} a^{13} + \frac{1}{12} a^{12} + \frac{7}{20} a^{11} + \frac{3}{10} a^{10} - \frac{13}{30} a^{9} + \frac{1}{15} a^{8} - \frac{1}{60} a^{7} - \frac{11}{30} a^{6} - \frac{7}{30} a^{5} - \frac{4}{15} a^{4} - \frac{1}{4} a^{3} - \frac{1}{12} a^{2} - \frac{1}{3} a + \frac{5}{12}$, $\frac{1}{1931880372678271310710020} a^{15} + \frac{26816834281504590359}{3389263811716265457386} a^{14} - \frac{56432522608424025135331}{386376074535654262142004} a^{13} + \frac{278930040126854667638101}{1931880372678271310710020} a^{12} - \frac{65981622994969908564247}{321980062113045218451670} a^{11} - \frac{127269687265929591863833}{965940186339135655355010} a^{10} + \frac{44662775158721154511972}{160990031056522609225835} a^{9} + \frac{681478465825224389152099}{1931880372678271310710020} a^{8} - \frac{2904953452999564759723}{16946319058581327286930} a^{7} + \frac{385798821789474401567293}{965940186339135655355010} a^{6} + \frac{173245469841102643401706}{482970093169567827677505} a^{5} - \frac{69340717190721613642459}{386376074535654262142004} a^{4} - \frac{73904269133743148935309}{386376074535654262142004} a^{3} - \frac{657140032989284555434}{1694631905858132728693} a^{2} - \frac{54064029791610159976977}{128792024845218087380668} a - \frac{19219267156755567147983}{96594018633913565535501}$

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:  \( 11574471.2198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1188:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 52 conjugacy class representatives for t16n1188 are not computed
Character table for t16n1188 is not computed

Intermediate fields

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 8.4.141288050000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.16.19$x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 12 x^{2} + 12$$4$$2$$16$$C_2^3: C_4$$[2, 3, 3]^{4}$
$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}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
41Data not computed