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Properties

Label 20.0.59713087499...5376.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $21.82$
Ramified primes $2, 7, 17$
Class number $4$
Class group $[4]$
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 0, -175, 0, 592, 0, -472, 0, 114, 0, 144, 0, -104, 0, 9, 0, 18, 0, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 + 18*x^16 + 9*x^14 - 104*x^12 + 144*x^10 + 114*x^8 - 472*x^6 + 592*x^4 - 175*x^2 + 49)
 
gp: K = bnfinit(x^20 - 7*x^18 + 18*x^16 + 9*x^14 - 104*x^12 + 144*x^10 + 114*x^8 - 472*x^6 + 592*x^4 - 175*x^2 + 49, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{18} + 18 x^{16} + 9 x^{14} - 104 x^{12} + 144 x^{10} + 114 x^{8} - 472 x^{6} + 592 x^{4} - 175 x^{2} + 49 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(597130874990690290159845376=2^{20}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.82$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{2}{7} a^{8} + \frac{3}{7} a^{6} + \frac{2}{7} a^{4} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{11} + \frac{2}{7} a^{9} + \frac{3}{7} a^{7} + \frac{2}{7} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{8} + \frac{3}{7} a^{6} - \frac{3}{7} a^{4} - \frac{2}{7} a^{2}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{9} + \frac{3}{7} a^{7} - \frac{3}{7} a^{5} - \frac{2}{7} a^{3}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{49} a^{15} + \frac{1}{49} a^{13} + \frac{3}{49} a^{11} + \frac{17}{49} a^{9} + \frac{19}{49} a^{7} + \frac{24}{49} a^{5} - \frac{12}{49} a^{3} - \frac{1}{7} a$, $\frac{1}{833} a^{16} - \frac{55}{833} a^{14} + \frac{59}{833} a^{12} + \frac{59}{833} a^{10} + \frac{208}{833} a^{8} - \frac{270}{833} a^{6} + \frac{247}{833} a^{4} - \frac{12}{119} a^{2} + \frac{8}{17}$, $\frac{1}{833} a^{17} - \frac{4}{833} a^{15} - \frac{9}{833} a^{13} - \frac{26}{833} a^{11} - \frac{115}{833} a^{9} - \frac{372}{833} a^{7} - \frac{314}{833} a^{5} + \frac{137}{833} a^{3} + \frac{5}{119} a$, $\frac{1}{9930193} a^{18} - \frac{773}{9930193} a^{16} + \frac{687504}{9930193} a^{14} + \frac{101330}{9930193} a^{12} + \frac{1039}{28951} a^{10} - \frac{661314}{9930193} a^{8} + \frac{2940151}{9930193} a^{6} + \frac{364496}{9930193} a^{4} + \frac{159853}{1418599} a^{2} - \frac{53769}{202657}$, $\frac{1}{9930193} a^{19} - \frac{773}{9930193} a^{17} + \frac{79533}{9930193} a^{15} - \frac{506641}{9930193} a^{13} - \frac{6991}{1418599} a^{11} + \frac{1770570}{9930193} a^{9} - \frac{4355501}{9930193} a^{7} - \frac{1459417}{9930193} a^{5} - \frac{1979}{202657} a^{3} + \frac{33084}{202657} a$

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

Class group and class number

$C_{4}$, 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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 83925.1811566 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

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

Intermediate fields

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{7}, \sqrt{-17})\), 5.1.14161.1 x5, 10.0.23863536599.2, 10.0.3490894496768.4 x5, 10.2.1437427145728.4 x5

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

Sibling fields

Degree 10 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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$