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Properties

Label 10.10.2616685559...3125.1
Degree $10$
Signature $[10, 0]$
Discriminant $5^{13}\cdot 11^{8}$
Root discriminant $55.18$
Ramified primes $5, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_5$ (as 10T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-980, 0, 7725, 0, -4265, 0, 700, 0, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 45*x^8 + 700*x^6 - 4265*x^4 + 7725*x^2 - 980)
 
gp: K = bnfinit(x^10 - 45*x^8 + 700*x^6 - 4265*x^4 + 7725*x^2 - 980, 1)
 

Normalized defining polynomial

\( x^{10} - 45 x^{8} + 700 x^{6} - 4265 x^{4} + 7725 x^{2} - 980 \)

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

Invariants

Degree:  $10$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(261668555908203125=5^{13}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.18$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{84} a^{7} - \frac{1}{42} a^{5} - \frac{1}{42} a^{3} - \frac{1}{2} a^{2} + \frac{31}{84} a - \frac{1}{2}$, $\frac{1}{17136} a^{8} - \frac{235}{4284} a^{6} + \frac{146}{1071} a^{4} - \frac{1}{2} a^{3} + \frac{4175}{17136} a^{2} - \frac{1}{2} a - \frac{65}{612}$, $\frac{1}{34272} a^{9} - \frac{1}{34272} a^{8} - \frac{31}{8568} a^{7} - \frac{479}{8568} a^{6} - \frac{983}{4284} a^{5} - \frac{649}{4284} a^{4} - \frac{14593}{34272} a^{3} + \frac{1537}{34272} a^{2} + \frac{3727}{8568} a - \frac{343}{1224}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 280688.94507 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5$ (as 10T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10
The 4 conjugacy class representatives for $D_5$
Character table for $D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.228765625.1 x5

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

Sibling fields

Degree 5 sibling: 5.5.228765625.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$5.10.13.1$x^{10} + 15 x^{4} + 5$$10$$1$$13$$D_5$$[3/2]_{2}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$