Properties

Label 10.0.534284452043128832.2
Degree $10$
Signature $[0, 5]$
Discriminant $-5.343\times 10^{17}$
Root discriminant $59.26$
Ramified primes $2, 439$
Class number $20$
Class group $[20]$
Galois group $D_{10}$ (as 10T3)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 28*x^8 + 556*x^6 - 1528*x^4 + 18352*x^2 + 14048)
 
gp: K = bnfinit(x^10 - 28*x^8 + 556*x^6 - 1528*x^4 + 18352*x^2 + 14048, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14048, 0, 18352, 0, -1528, 0, 556, 0, -28, 0, 1]);
 

\(x^{10} - 28 x^{8} + 556 x^{6} - 1528 x^{4} + 18352 x^{2} + 14048\)  Toggle raw display

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

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-534284452043128832\)\(\medspace = -\,2^{15}\cdot 439^{5}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $59.26$
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:  $2, 439$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{12} a^{4} - \frac{1}{3}$, $\frac{1}{12} a^{5} - \frac{1}{3} a$, $\frac{1}{24} a^{6} - \frac{1}{6} a^{2}$, $\frac{1}{24} a^{7} - \frac{1}{6} a^{3}$, $\frac{1}{7751952} a^{8} - \frac{19}{8282} a^{6} - \frac{7424}{484497} a^{4} + \frac{637}{8282} a^{2} + \frac{155587}{484497}$, $\frac{1}{7751952} a^{9} - \frac{19}{8282} a^{7} - \frac{7424}{484497} a^{5} + \frac{637}{8282} a^{3} + \frac{155587}{484497} a$  Toggle raw display

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

Class group and class number

$C_{20}$, which has order $20$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5257.98433890173 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{5}\cdot 5257.98433890173 \cdot 20}{2\sqrt{534284452043128832}}\approx 0.704421109262057$

Galois group

$D_{10}$ (as 10T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-878}) \), 5.1.192721.1

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

Sibling fields

Galois closure: 20.0.285459875695026432742900224149684224.2
Degree 10 sibling: 10.2.1217048865701888.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.2.0.1}{2} }^{5}$ ${\href{/padicField/17.2.0.1}{2} }^{5}$ ${\href{/padicField/19.10.0.1}{10} }$ ${\href{/padicField/23.2.0.1}{2} }^{5}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.2.0.1}{2} }^{5}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{5}$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{5}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
439Data not computed