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Properties

Label 16.0.26436931289...0625.7
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{8}$
Root discriminant $33.60$
Ramified primes $5, 101$
Class number $32$ (GRH)
Class group $[2, 2, 2, 4]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 0, 1375, 0, 2900, 0, 6105, 0, 12851, 0, 1221, 0, 116, 0, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 11*x^14 + 116*x^12 + 1221*x^10 + 12851*x^8 + 6105*x^6 + 2900*x^4 + 1375*x^2 + 625)
 
gp: K = bnfinit(x^16 + 11*x^14 + 116*x^12 + 1221*x^10 + 12851*x^8 + 6105*x^6 + 2900*x^4 + 1375*x^2 + 625, 1)
 

Normalized defining polynomial

\( x^{16} + 11 x^{14} + 116 x^{12} + 1221 x^{10} + 12851 x^{8} + 6105 x^{6} + 2900 x^{4} + 1375 x^{2} + 625 \)

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:  \(2643693128974804931640625=5^{12}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.60$
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 Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{128510} a^{10} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{5815}{12851}$, $\frac{1}{128510} a^{11} + \frac{1}{10} a^{9} + \frac{1}{10} a^{7} - \frac{2}{5} a^{5} + \frac{1}{10} a^{3} - \frac{1}{2} a^{2} + \frac{1221}{25702} a - \frac{1}{2}$, $\frac{1}{642550} a^{12} + \frac{1}{642550} a^{10} + \frac{3}{25} a^{8} - \frac{1}{2} a^{7} - \frac{7}{25} a^{6} - \frac{1}{2} a^{5} + \frac{8}{25} a^{4} - \frac{24481}{128510} a^{2} + \frac{10525}{25702}$, $\frac{1}{3212750} a^{13} - \frac{2}{1606375} a^{11} + \frac{51}{250} a^{9} - \frac{22}{125} a^{7} - \frac{1}{2} a^{6} + \frac{11}{250} a^{5} - \frac{1}{2} a^{4} + \frac{155433}{642550} a^{3} - \frac{1}{2} a^{2} - \frac{29249}{128510} a$, $\frac{1}{3212750} a^{14} + \frac{1}{3212750} a^{12} + \frac{3}{1606375} a^{10} - \frac{39}{250} a^{8} - \frac{1}{2} a^{7} + \frac{41}{250} a^{6} - \frac{1}{2} a^{5} + \frac{148397}{321275} a^{4} - \frac{1163}{64255} a^{2} - \frac{1}{2} a + \frac{6315}{12851}$, $\frac{1}{16063750} a^{15} + \frac{1}{16063750} a^{13} + \frac{3}{8031875} a^{11} + \frac{211}{1250} a^{9} - \frac{167}{625} a^{7} - \frac{1}{2} a^{6} + \frac{618069}{3212750} a^{5} - \frac{1}{2} a^{4} - \frac{195091}{642550} a^{3} - \frac{1}{2} a^{2} + \frac{25481}{128510} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (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:  \( -\frac{1276}{1606375} a^{14} - \frac{13456}{1606375} a^{12} - \frac{141636}{1606375} a^{10} - \frac{116}{125} a^{8} - \frac{1221}{125} a^{6} - \frac{13456}{64255} a^{4} - \frac{1276}{12851} a^{2} - \frac{580}{12851} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 103071.243274 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{505}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.4.51005.1 x2, 4.4.2525.1 x2, 4.0.1275125.1 x2, 4.0.12625.1 x2, \(\Q(\zeta_{5})\), 4.0.1275125.2, 8.8.65037750625.1, 8.0.1625943765625.5, 8.0.1625943765625.6, 8.0.159390625.1 x2, 8.0.1625943765625.4 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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$101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$