Properties

Label 20.0.33545186845...0752.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 13^{15}$
Root discriminant $11.92$
Ramified primes $2, 13$
Class number $1$
Class group Trivial
Galois Group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 0, 10, -15, 0, 40, -64, 46, 8, -32, 8, 46, -64, 40, 0, -15, 10, 0, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 10*x^17 - 15*x^16 + 40*x^14 - 64*x^13 + 46*x^12 + 8*x^11 - 32*x^10 + 8*x^9 + 46*x^8 - 64*x^7 + 40*x^6 - 15*x^4 + 10*x^3 - 2*x + 1)
gp: K = bnfinit(x^20 - 2*x^19 + 10*x^17 - 15*x^16 + 40*x^14 - 64*x^13 + 46*x^12 + 8*x^11 - 32*x^10 + 8*x^9 + 46*x^8 - 64*x^7 + 40*x^6 - 15*x^4 + 10*x^3 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 2 x^{19} \) \(\mathstrut +\mathstrut 10 x^{17} \) \(\mathstrut -\mathstrut 15 x^{16} \) \(\mathstrut +\mathstrut 40 x^{14} \) \(\mathstrut -\mathstrut 64 x^{13} \) \(\mathstrut +\mathstrut 46 x^{12} \) \(\mathstrut +\mathstrut 8 x^{11} \) \(\mathstrut -\mathstrut 32 x^{10} \) \(\mathstrut +\mathstrut 8 x^{9} \) \(\mathstrut +\mathstrut 46 x^{8} \) \(\mathstrut -\mathstrut 64 x^{7} \) \(\mathstrut +\mathstrut 40 x^{6} \) \(\mathstrut -\mathstrut 15 x^{4} \) \(\mathstrut +\mathstrut 10 x^{3} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3354518684571451850752=2^{16}\cdot 13^{15}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.92$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 13$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{8} + \frac{1}{8}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{9} + \frac{1}{8} a$, $\frac{1}{8} a^{18} - \frac{1}{4} a^{8} - \frac{1}{8} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{19} - \frac{1}{16} a^{18} - \frac{1}{16} a^{17} - \frac{1}{16} a^{16} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{4} + \frac{1}{16} a^{3} + \frac{3}{16} a^{2} - \frac{1}{16} a - \frac{5}{16}$

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

Class group and class number

Trivial Abelian group, 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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{9}{8} a^{19} - \frac{1}{4} a^{18} - \frac{21}{8} a^{17} + \frac{35}{4} a^{16} + \frac{9}{4} a^{15} - 16 a^{14} + \frac{55}{2} a^{13} + a^{12} - \frac{81}{4} a^{11} + 27 a^{10} + \frac{31}{2} a^{9} - 26 a^{8} + \frac{109}{4} a^{7} + 12 a^{6} - 16 a^{5} + 9 a^{4} + \frac{69}{8} a^{3} - \frac{19}{4} a^{2} + \frac{5}{8} a + \frac{9}{4} \),  \( \frac{5}{8} a^{19} - a^{18} - \frac{3}{4} a^{17} + \frac{13}{2} a^{16} - \frac{51}{8} a^{15} - \frac{49}{8} a^{14} + \frac{209}{8} a^{13} - \frac{211}{8} a^{12} + \frac{31}{8} a^{11} + \frac{173}{8} a^{10} - \frac{115}{8} a^{9} - \frac{97}{8} a^{8} + \frac{259}{8} a^{7} - \frac{171}{8} a^{6} - \frac{9}{8} a^{5} + \frac{95}{8} a^{4} - \frac{9}{2} a^{3} - \frac{17}{8} a^{2} + \frac{25}{8} a + \frac{1}{8} \),  \( \frac{1}{2} a^{19} - a^{18} + \frac{11}{2} a^{16} - 8 a^{15} - a^{14} + \frac{49}{2} a^{13} - 34 a^{12} + \frac{33}{2} a^{11} + \frac{39}{2} a^{10} - 26 a^{9} + \frac{3}{2} a^{8} + \frac{69}{2} a^{7} - 34 a^{6} + \frac{21}{2} a^{5} + \frac{31}{2} a^{4} - \frac{29}{2} a^{3} + \frac{5}{2} a^{2} + 4 a - \frac{3}{2} \),  \( a^{19} - \frac{1}{2} a^{18} - \frac{9}{4} a^{17} + \frac{67}{8} a^{16} - \frac{1}{8} a^{15} - \frac{115}{8} a^{14} + \frac{223}{8} a^{13} - \frac{49}{8} a^{12} - \frac{127}{8} a^{11} + \frac{219}{8} a^{10} + \frac{49}{8} a^{9} - \frac{185}{8} a^{8} + \frac{237}{8} a^{7} + \frac{15}{8} a^{6} - \frac{111}{8} a^{5} + \frac{97}{8} a^{4} + \frac{27}{8} a^{3} - \frac{35}{8} a^{2} + \frac{17}{8} a + \frac{3}{4} \),  \( \frac{3}{4} a^{19} - \frac{5}{2} a^{18} + \frac{1}{2} a^{17} + \frac{39}{4} a^{16} - \frac{157}{8} a^{15} + \frac{1}{8} a^{14} + \frac{355}{8} a^{13} - \frac{607}{8} a^{12} + \frac{325}{8} a^{11} + \frac{175}{8} a^{10} - \frac{411}{8} a^{9} - \frac{1}{8} a^{8} + \frac{461}{8} a^{7} - \frac{621}{8} a^{6} + \frac{225}{8} a^{5} + \frac{111}{8} a^{4} - \frac{187}{8} a^{3} + \frac{33}{8} a^{2} + \frac{35}{8} a - \frac{29}{8} \),  \( \frac{17}{16} a^{19} - \frac{33}{16} a^{18} - \frac{11}{16} a^{17} + \frac{173}{16} a^{16} - \frac{57}{4} a^{15} - \frac{21}{4} a^{14} + \frac{85}{2} a^{13} - \frac{119}{2} a^{12} + \frac{245}{8} a^{11} + \frac{133}{8} a^{10} - \frac{241}{8} a^{9} - \frac{47}{8} a^{8} + \frac{191}{4} a^{7} - \frac{229}{4} a^{6} + \frac{47}{2} a^{5} + 4 a^{4} - \frac{211}{16} a^{3} + \frac{79}{16} a^{2} + \frac{13}{16} a - \frac{23}{16} \),  \( a^{19} - a^{18} - a^{17} + \frac{17}{2} a^{16} - \frac{51}{8} a^{15} - \frac{21}{4} a^{14} + \frac{249}{8} a^{13} - \frac{135}{4} a^{12} + \frac{147}{8} a^{11} + 16 a^{10} - \frac{119}{8} a^{9} - \frac{11}{4} a^{8} + \frac{295}{8} a^{7} - \frac{131}{4} a^{6} + \frac{119}{8} a^{5} + \frac{23}{4} a^{4} - \frac{79}{8} a^{3} + 3 a^{2} + \frac{15}{8} a - \frac{7}{4} \),  \( \frac{11}{16} a^{19} - \frac{9}{16} a^{18} - \frac{17}{16} a^{17} + \frac{93}{16} a^{16} - \frac{5}{2} a^{15} - \frac{25}{4} a^{14} + 20 a^{13} - \frac{29}{2} a^{12} + \frac{31}{8} a^{11} + \frac{97}{8} a^{10} - \frac{13}{8} a^{9} - \frac{55}{8} a^{8} + 22 a^{7} - \frac{41}{4} a^{6} + 4 a^{5} + 4 a^{4} + \frac{15}{16} a^{3} + \frac{15}{16} a^{2} + \frac{11}{16} a + \frac{9}{16} \),  \( \frac{1}{8} a^{19} - \frac{1}{4} a^{18} + \frac{3}{8} a^{17} + a^{16} - \frac{19}{8} a^{15} + 3 a^{14} + \frac{31}{8} a^{13} - \frac{43}{4} a^{12} + \frac{127}{8} a^{11} - \frac{25}{4} a^{10} - \frac{5}{8} a^{9} + \frac{21}{4} a^{8} + \frac{59}{8} a^{7} - \frac{23}{2} a^{6} + \frac{125}{8} a^{5} - \frac{19}{4} a^{4} + 2 a^{3} + a^{2} + \frac{3}{4} a + \frac{1}{4} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 238.906276128 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 5.1.35152.1 x5, 10.2.16063620352.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.35152.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
2Data not computed
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$