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Properties

Label 20.0.14091984613...2224.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 103^{10}$
Root discriminant $20.30$
Ramified primes $2, 103$
Class number $1$
Class group Trivial
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![81, 0, 135, 0, 1021, 0, 1393, 0, 763, 0, 300, 0, 129, 0, -15, 0, -17, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 - 17*x^16 - 15*x^14 + 129*x^12 + 300*x^10 + 763*x^8 + 1393*x^6 + 1021*x^4 + 135*x^2 + 81)
 
gp: K = bnfinit(x^20 - 2*x^18 - 17*x^16 - 15*x^14 + 129*x^12 + 300*x^10 + 763*x^8 + 1393*x^6 + 1021*x^4 + 135*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{18} - 17 x^{16} - 15 x^{14} + 129 x^{12} + 300 x^{10} + 763 x^{8} + 1393 x^{6} + 1021 x^{4} + 135 x^{2} + 81 \)

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:  \(140919846138714198689972224=2^{20}\cdot 103^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.30$
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:  $2, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{10} + \frac{2}{5} a^{6} + \frac{1}{3} a^{4} + \frac{4}{15} a^{2} + \frac{2}{5}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{11} + \frac{1}{15} a^{7} - \frac{1}{3} a^{5} - \frac{1}{15} a^{3} - \frac{4}{15} a$, $\frac{1}{45} a^{14} + \frac{1}{45} a^{12} + \frac{1}{15} a^{10} - \frac{4}{45} a^{8} + \frac{4}{15} a^{6} + \frac{4}{45} a^{4} + \frac{19}{45} a^{2} - \frac{2}{5}$, $\frac{1}{45} a^{15} + \frac{1}{45} a^{13} + \frac{1}{15} a^{11} - \frac{4}{45} a^{9} - \frac{1}{15} a^{7} + \frac{19}{45} a^{5} + \frac{4}{45} a^{3} - \frac{1}{15} a$, $\frac{1}{225} a^{16} + \frac{1}{225} a^{14} - \frac{1}{75} a^{12} - \frac{13}{225} a^{10} - \frac{2}{25} a^{8} - \frac{47}{225} a^{6} + \frac{49}{225} a^{4} + \frac{16}{75} a^{2} + \frac{6}{25}$, $\frac{1}{225} a^{17} + \frac{1}{225} a^{15} - \frac{1}{75} a^{13} - \frac{13}{225} a^{11} - \frac{2}{25} a^{9} + \frac{28}{225} a^{7} - \frac{26}{225} a^{5} - \frac{34}{75} a^{3} - \frac{7}{75} a$, $\frac{1}{4050087525} a^{18} + \frac{355796}{4050087525} a^{16} + \frac{30770132}{4050087525} a^{14} + \frac{759243}{450009725} a^{12} + \frac{335735072}{4050087525} a^{10} + \frac{153227561}{1350029175} a^{8} + \frac{988757009}{4050087525} a^{6} + \frac{5738704}{86172075} a^{4} - \frac{303809846}{4050087525} a^{2} - \frac{26969156}{90001945}$, $\frac{1}{12150262575} a^{19} + \frac{3671237}{2430052515} a^{17} + \frac{48770521}{12150262575} a^{15} + \frac{14855857}{810017505} a^{13} - \frac{33740111}{270005835} a^{11} - \frac{134928166}{1350029175} a^{9} - \frac{937284614}{12150262575} a^{7} + \frac{110677142}{258516225} a^{5} + \frac{290202991}{12150262575} a^{3} - \frac{296849281}{1350029175} a$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{48403}{12912075} a^{19} - \frac{18376}{2582415} a^{17} - \frac{828122}{12912075} a^{15} - \frac{54259}{860805} a^{13} + \frac{8983}{19129} a^{11} + \frac{5052166}{4304025} a^{9} + \frac{38816398}{12912075} a^{7} + \frac{1543631}{274725} a^{5} + \frac{60732598}{12912075} a^{3} + \frac{2636702}{1434675} a \) (order $4$)
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:  \( 276624.618184 \)
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

\(\Q(\sqrt{-103}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{103}) \), \(\Q(i, \sqrt{103})\), 5.1.10609.1 x5, 10.0.11592740743.1, 10.0.115252102144.4 x5, 10.2.11870966520832.4 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
$103$103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$