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Properties

Label 16.0.60479166614...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{8}\cdot 7841^{4}$
Root discriminant $35.39$
Ramified primes $2, 5, 7841$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4273025, -867030, -2070881, 68274, 448796, 118200, -9170, -13266, 1390, 5130, 888, -570, 6, 64, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 64*x^13 + 6*x^12 - 570*x^11 + 888*x^10 + 5130*x^9 + 1390*x^8 - 13266*x^7 - 9170*x^6 + 118200*x^5 + 448796*x^4 + 68274*x^3 - 2070881*x^2 - 867030*x + 4273025)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 64*x^13 + 6*x^12 - 570*x^11 + 888*x^10 + 5130*x^9 + 1390*x^8 - 13266*x^7 - 9170*x^6 + 118200*x^5 + 448796*x^4 + 68274*x^3 - 2070881*x^2 - 867030*x + 4273025, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 64 x^{13} + 6 x^{12} - 570 x^{11} + 888 x^{10} + 5130 x^{9} + 1390 x^{8} - 13266 x^{7} - 9170 x^{6} + 118200 x^{5} + 448796 x^{4} + 68274 x^{3} - 2070881 x^{2} - 867030 x + 4273025 \)

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:  \(6047916661441537600000000=2^{12}\cdot 5^{8}\cdot 7841^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.39$
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:  $2, 5, 7841$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{386653120968720595914287265881923329799065061340} a^{15} + \frac{7380863592728879031660006145498738435008490924}{96663280242180148978571816470480832449766265335} a^{14} - \frac{70673825651083882762023160617651174783723006823}{386653120968720595914287265881923329799065061340} a^{13} - \frac{1602558094200062561819801110335783651253893033}{55236160138388656559183895125989047114152151620} a^{12} + \frac{34244893282048744880839014916523820653859678704}{96663280242180148978571816470480832449766265335} a^{11} - \frac{7499291340225119308692396000797560688199388437}{77330624193744119182857453176384665959813012268} a^{10} + \frac{53536633986376739901780182151047703028696305603}{386653120968720595914287265881923329799065061340} a^{9} - \frac{2976003673488429618966211195642794859790724398}{19332656048436029795714363294096166489953253067} a^{8} - \frac{2783448608160621740342542738203090449424931127}{77330624193744119182857453176384665959813012268} a^{7} + \frac{168753180291846739617701222648509291009375877019}{386653120968720595914287265881923329799065061340} a^{6} - \frac{5773118743636560001673743371273985292543678867}{19332656048436029795714363294096166489953253067} a^{5} + \frac{170769257057772568701513327624925718919241811}{77330624193744119182857453176384665959813012268} a^{4} - \frac{21958310829955645997201938208787944470635752857}{55236160138388656559183895125989047114152151620} a^{3} + \frac{16742275975629757822623640275220083365080307}{193326560484360297957143632940961664899532530670} a^{2} - \frac{33496492315118314095630703767485187519747951233}{193326560484360297957143632940961664899532530670} a + \frac{15194507302666597612719033502551859172161106501}{77330624193744119182857453176384665959813012268}$

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:  $7$
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:  \( 1358570.75849 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.0.4900625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.9$x^{12} - 18 x^{10} + 7 x^{8} - 28 x^{6} - x^{4} - 18 x^{2} - 7$$2$$6$$12$12T58$[2, 2, 2, 2]^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7841Data not computed