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Properties

Label 16.0.34336147906...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $19.21$
Ramified primes $3, 5, 11$
Class number $4$
Class group $[2, 2]$
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![1, -4, 20, 123, 631, 496, 447, -28, 453, 113, 162, -71, 76, -3, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 5*x^14 - 3*x^13 + 76*x^12 - 71*x^11 + 162*x^10 + 113*x^9 + 453*x^8 - 28*x^7 + 447*x^6 + 496*x^5 + 631*x^4 + 123*x^3 + 20*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 5*x^14 - 3*x^13 + 76*x^12 - 71*x^11 + 162*x^10 + 113*x^9 + 453*x^8 - 28*x^7 + 447*x^6 + 496*x^5 + 631*x^4 + 123*x^3 + 20*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 5 x^{14} - 3 x^{13} + 76 x^{12} - 71 x^{11} + 162 x^{10} + 113 x^{9} + 453 x^{8} - 28 x^{7} + 447 x^{6} + 496 x^{5} + 631 x^{4} + 123 x^{3} + 20 x^{2} - 4 x + 1 \)

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:  \(343361479062744140625=3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.21$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{10} a^{7} + \frac{3}{10} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{10} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{10}$, $\frac{1}{22234740} a^{14} + \frac{39769}{5558685} a^{13} - \frac{34657}{1852895} a^{12} - \frac{592181}{4446948} a^{11} + \frac{151289}{22234740} a^{10} + \frac{676449}{1482316} a^{9} - \frac{980918}{5558685} a^{8} + \frac{769799}{11117370} a^{7} - \frac{128641}{2021340} a^{6} + \frac{1889827}{7411580} a^{5} + \frac{8746703}{22234740} a^{4} - \frac{4706291}{11117370} a^{3} - \frac{21697}{741158} a^{2} - \frac{485003}{22234740} a - \frac{4564907}{22234740}$, $\frac{1}{1136084040300} a^{15} - \frac{6187}{284021010075} a^{14} - \frac{1187318581}{25820091825} a^{13} + \frac{4968378289}{227216808060} a^{12} + \frac{227880754861}{1136084040300} a^{11} + \frac{13071496667}{1136084040300} a^{10} - \frac{1881761513}{284021010075} a^{9} + \frac{39312412501}{568042020150} a^{8} - \frac{27977339581}{1136084040300} a^{7} + \frac{383564387089}{1136084040300} a^{6} + \frac{299355109259}{1136084040300} a^{5} + \frac{44588251979}{568042020150} a^{4} - \frac{4376433959}{22721680806} a^{3} - \frac{509504031677}{1136084040300} a^{2} - \frac{30040913261}{103280367300} a - \frac{92618715278}{284021010075}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1264.83496836 \)
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

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.2.275.1 x2, 4.0.605.1 x2, 4.0.136125.1 x2, 4.2.12375.1 x2, \(\Q(\zeta_{15})^+\), 4.0.136125.2, 8.0.9150625.1, 8.0.18530015625.6, 8.0.18530015625.1, 8.4.153140625.2 x2, 8.0.18530015625.8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$