Properties

Label 16.0.34297420960...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $22.18$
Ramified primes $2, 5, 11$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^14 + 4*x^12 + 3*x^10 - 29*x^8 + 15*x^6 + 100*x^4 - 375*x^2 + 625)
 
gp: K = bnfinit(x^16 - 3*x^14 + 4*x^12 + 3*x^10 - 29*x^8 + 15*x^6 + 100*x^4 - 375*x^2 + 625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 0, -375, 0, 100, 0, 15, 0, -29, 0, 3, 0, 4, 0, -3, 0, 1]);
 

Normalized defining polynomial

\( x^{16} - 3 x^{14} + 4 x^{12} + 3 x^{10} - 29 x^{8} + 15 x^{6} + 100 x^{4} - 375 x^{2} + 625 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3429742096000000000000=2^{16}\cdot 5^{12}\cdot 11^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.18$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $16$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{145} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} + \frac{3}{29}$, $\frac{1}{145} a^{11} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{3}{29} a$, $\frac{1}{725} a^{12} + \frac{2}{725} a^{10} - \frac{9}{25} a^{8} + \frac{12}{25} a^{6} + \frac{9}{25} a^{4} - \frac{26}{145} a^{2} + \frac{7}{29}$, $\frac{1}{3625} a^{13} + \frac{2}{3625} a^{11} - \frac{59}{125} a^{9} - \frac{13}{125} a^{7} - \frac{41}{125} a^{5} - \frac{171}{725} a^{3} - \frac{51}{145} a$, $\frac{1}{3625} a^{14} + \frac{2}{3625} a^{12} - \frac{11}{3625} a^{10} - \frac{38}{125} a^{8} + \frac{34}{125} a^{6} - \frac{26}{725} a^{4} + \frac{7}{145} a^{2} + \frac{1}{29}$, $\frac{1}{18125} a^{15} + \frac{2}{18125} a^{13} + \frac{39}{18125} a^{11} - \frac{13}{625} a^{9} - \frac{291}{625} a^{7} + \frac{554}{3625} a^{5} + \frac{94}{725} a^{3} + \frac{13}{29} a$

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

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{1}{145} a^{12} + \frac{72}{145} a^{2} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 29630.7767341 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.0.9680.1 x2, 4.2.4400.1 x2, 4.0.242000.1 x2, 4.2.22000.1 x2, 4.4.15125.1, \(\Q(\zeta_{5})\), 8.0.2342560000.4, 8.0.58564000000.6, 8.0.228765625.1, 8.4.58564000000.1 x2, 8.0.484000000.5 x2

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

Sibling fields

Degree 8 siblings: 8.4.58564000000.1, 8.0.484000000.5

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$