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Properties

Label 16.0.25916019301...5625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{6}$
Root discriminant $18.87$
Ramified primes $5, 101$
Class number $2$
Class group $[2]$
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 100, 315, 505, 416, 133, -75, 80, 11, -275, 246, -200, 101, -40, 15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 15*x^14 - 40*x^13 + 101*x^12 - 200*x^11 + 246*x^10 - 275*x^9 + 11*x^8 + 80*x^7 - 75*x^6 + 133*x^5 + 416*x^4 + 505*x^3 + 315*x^2 + 100*x + 25)
 
gp: K = bnfinit(x^16 - 3*x^15 + 15*x^14 - 40*x^13 + 101*x^12 - 200*x^11 + 246*x^10 - 275*x^9 + 11*x^8 + 80*x^7 - 75*x^6 + 133*x^5 + 416*x^4 + 505*x^3 + 315*x^2 + 100*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 15 x^{14} - 40 x^{13} + 101 x^{12} - 200 x^{11} + 246 x^{10} - 275 x^{9} + 11 x^{8} + 80 x^{7} - 75 x^{6} + 133 x^{5} + 416 x^{4} + 505 x^{3} + 315 x^{2} + 100 x + 25 \)

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:  \(259160193017822265625=5^{12}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.87$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{355} a^{14} + \frac{102}{355} a^{13} - \frac{12}{71} a^{12} + \frac{25}{71} a^{11} + \frac{26}{355} a^{10} - \frac{29}{71} a^{9} - \frac{29}{355} a^{8} - \frac{15}{71} a^{7} - \frac{44}{355} a^{6} - \frac{19}{71} a^{5} + \frac{30}{71} a^{4} - \frac{47}{355} a^{3} + \frac{81}{355} a^{2} + \frac{18}{71} a - \frac{21}{71}$, $\frac{1}{63455350210255} a^{15} + \frac{29100111023}{63455350210255} a^{14} + \frac{23365291115337}{63455350210255} a^{13} - \frac{1017251845452}{12691070042051} a^{12} + \frac{8152918038986}{63455350210255} a^{11} - \frac{17499017894664}{63455350210255} a^{10} - \frac{206331926639}{893737326905} a^{9} - \frac{14124226277354}{63455350210255} a^{8} + \frac{1678485055506}{63455350210255} a^{7} + \frac{5977590450941}{63455350210255} a^{6} + \frac{1787824057891}{12691070042051} a^{5} - \frac{2109481417202}{63455350210255} a^{4} - \frac{11862174432936}{63455350210255} a^{3} + \frac{7376005860026}{63455350210255} a^{2} + \frac{5057063068986}{12691070042051} a - \frac{2592794518586}{12691070042051}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{2119251492813}{63455350210255} a^{15} - \frac{5792468990947}{63455350210255} a^{14} + \frac{28706103897549}{63455350210255} a^{13} - \frac{14736070372368}{12691070042051} a^{12} + \frac{175881114936818}{63455350210255} a^{11} - \frac{4651570963478}{893737326905} a^{10} + \frac{326444122467393}{63455350210255} a^{9} - \frac{283743905123003}{63455350210255} a^{8} - \frac{258534647751207}{63455350210255} a^{7} + \frac{367445787773607}{63455350210255} a^{6} - \frac{22037624309386}{12691070042051} a^{5} + \frac{215037656257939}{63455350210255} a^{4} + \frac{808861959591139}{63455350210255} a^{3} + \frac{932216226222152}{63455350210255} a^{2} + \frac{92021038988098}{12691070042051} a + \frac{22586060686627}{12691070042051} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12699.1258424 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, \(\Q(\zeta_{5})\), 4.0.12625.1, 8.4.643938125.1, 8.4.16098453125.1, 8.0.159390625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.1$x^{4} - 101$$4$$1$$3$$C_4$$[\ ]_{4}$