Properties

Label 20.0.33010132986...7241.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 236438047^{2}$
Root discriminant $11.91$
Ramified primes $3, 236438047$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois Group 20T1021

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1, 0, -1, 3, -5, 5, -4, 4, 1, -5, 6, -10, 11, -9, 8, -6, 3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^2 - x + 1)
gp: K = bnfinit(x^20 - 2*x^19 + 3*x^18 - 6*x^17 + 8*x^16 - 9*x^15 + 11*x^14 - 10*x^13 + 6*x^12 - 5*x^11 + x^10 + 4*x^9 - 4*x^8 + 5*x^7 - 5*x^6 + 3*x^5 - x^4 + x^2 - x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 2 x^{19} \) \(\mathstrut +\mathstrut 3 x^{18} \) \(\mathstrut -\mathstrut 6 x^{17} \) \(\mathstrut +\mathstrut 8 x^{16} \) \(\mathstrut -\mathstrut 9 x^{15} \) \(\mathstrut +\mathstrut 11 x^{14} \) \(\mathstrut -\mathstrut 10 x^{13} \) \(\mathstrut +\mathstrut 6 x^{12} \) \(\mathstrut -\mathstrut 5 x^{11} \) \(\mathstrut +\mathstrut x^{10} \) \(\mathstrut +\mathstrut 4 x^{9} \) \(\mathstrut -\mathstrut 4 x^{8} \) \(\mathstrut +\mathstrut 5 x^{7} \) \(\mathstrut -\mathstrut 5 x^{6} \) \(\mathstrut +\mathstrut 3 x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut +\mathstrut x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(3301013298634667867241=3^{10}\cdot 236438047^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.91$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 236438047$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{59} a^{19} - \frac{26}{59} a^{18} - \frac{22}{59} a^{17} - \frac{9}{59} a^{16} - \frac{12}{59} a^{15} - \frac{16}{59} a^{14} - \frac{18}{59} a^{13} + \frac{9}{59} a^{12} + \frac{26}{59} a^{11} + \frac{20}{59} a^{10} - \frac{7}{59} a^{9} - \frac{5}{59} a^{8} - \frac{2}{59} a^{7} - \frac{6}{59} a^{6} + \frac{21}{59} a^{5} - \frac{29}{59} a^{4} - \frac{13}{59} a^{3} + \frac{17}{59} a^{2} + \frac{6}{59} a - \frac{27}{59}$

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

Class group and class number

Trivial Abelian group, order $1$ (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:  \( -\frac{13}{59} a^{19} + \frac{43}{59} a^{18} - \frac{68}{59} a^{17} + \frac{117}{59} a^{16} - \frac{198}{59} a^{15} + \frac{267}{59} a^{14} - \frac{297}{59} a^{13} + \frac{355}{59} a^{12} - \frac{338}{59} a^{11} + \frac{271}{59} a^{10} - \frac{263}{59} a^{9} + \frac{183}{59} a^{8} - \frac{33}{59} a^{7} + \frac{19}{59} a^{6} - \frac{37}{59} a^{5} - \frac{95}{59} a^{4} + \frac{51}{59} a^{3} - \frac{44}{59} a^{2} + \frac{40}{59} a - \frac{3}{59} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{17}{59} a^{19} - \frac{29}{59} a^{18} + \frac{39}{59} a^{17} - \frac{94}{59} a^{16} + \frac{150}{59} a^{15} - \frac{154}{59} a^{14} + \frac{225}{59} a^{13} - \frac{260}{59} a^{12} + \frac{206}{59} a^{11} - \frac{250}{59} a^{10} + \frac{235}{59} a^{9} - \frac{85}{59} a^{8} + \frac{84}{59} a^{7} - \frac{102}{59} a^{6} - \frac{56}{59} a^{5} + \frac{38}{59} a^{4} - \frac{44}{59} a^{3} + \frac{53}{59} a^{2} - \frac{75}{59} a + \frac{13}{59} \),  \( \frac{1}{59} a^{19} + \frac{33}{59} a^{18} - \frac{81}{59} a^{17} + \frac{109}{59} a^{16} - \frac{189}{59} a^{15} + \frac{279}{59} a^{14} - \frac{313}{59} a^{13} + \frac{363}{59} a^{12} - \frac{387}{59} a^{11} + \frac{256}{59} a^{10} - \frac{243}{59} a^{9} + \frac{231}{59} a^{8} - \frac{61}{59} a^{7} - \frac{6}{59} a^{6} + \frac{21}{59} a^{5} - \frac{88}{59} a^{4} + \frac{105}{59} a^{3} - \frac{42}{59} a^{2} + \frac{65}{59} a - \frac{27}{59} \),  \( \frac{39}{59} a^{19} - \frac{129}{59} a^{18} + \frac{204}{59} a^{17} - \frac{351}{59} a^{16} + \frac{535}{59} a^{15} - \frac{624}{59} a^{14} + \frac{714}{59} a^{13} - \frac{711}{59} a^{12} + \frac{483}{59} a^{11} - \frac{282}{59} a^{10} + \frac{140}{59} a^{9} + \frac{159}{59} a^{8} - \frac{255}{59} a^{7} + \frac{238}{59} a^{6} - \frac{243}{59} a^{5} + \frac{108}{59} a^{4} - \frac{35}{59} a^{3} - \frac{45}{59} a^{2} + \frac{57}{59} a - \frac{50}{59} \),  \( \frac{13}{59} a^{19} - \frac{43}{59} a^{18} + \frac{68}{59} a^{17} - \frac{117}{59} a^{16} + \frac{139}{59} a^{15} - \frac{149}{59} a^{14} + \frac{179}{59} a^{13} - \frac{119}{59} a^{12} + \frac{43}{59} a^{11} - \frac{35}{59} a^{10} - \frac{32}{59} a^{9} + \frac{53}{59} a^{8} + \frac{33}{59} a^{7} + \frac{40}{59} a^{6} - \frac{22}{59} a^{5} - \frac{82}{59} a^{4} + \frac{8}{59} a^{3} - \frac{15}{59} a^{2} + \frac{19}{59} a + \frac{3}{59} \),  \( \frac{53}{59} a^{19} - \frac{80}{59} a^{18} + \frac{73}{59} a^{17} - \frac{182}{59} a^{16} + \frac{190}{59} a^{15} - \frac{81}{59} a^{14} + \frac{108}{59} a^{13} + \frac{5}{59} a^{12} - \frac{274}{59} a^{11} + \frac{175}{59} a^{10} - \frac{194}{59} a^{9} + \frac{384}{59} a^{8} - \frac{106}{59} a^{7} - \frac{23}{59} a^{6} - \frac{67}{59} a^{5} - \frac{62}{59} a^{4} + \frac{137}{59} a^{3} - \frac{43}{59} a^{2} + \frac{23}{59} a - \frac{15}{59} \),  \( \frac{36}{59} a^{19} - \frac{51}{59} a^{18} + \frac{34}{59} a^{17} - \frac{147}{59} a^{16} + \frac{158}{59} a^{15} - \frac{104}{59} a^{14} + \frac{237}{59} a^{13} - \frac{207}{59} a^{12} + \frac{51}{59} a^{11} - \frac{224}{59} a^{10} + \frac{161}{59} a^{9} + \frac{115}{59} a^{8} + \frac{105}{59} a^{7} - \frac{39}{59} a^{6} - \frac{188}{59} a^{5} + \frac{18}{59} a^{4} + \frac{4}{59} a^{3} + \frac{81}{59} a^{2} - \frac{20}{59} a - \frac{28}{59} \),  \( \frac{14}{59} a^{19} - \frac{69}{59} a^{18} + \frac{105}{59} a^{17} - \frac{126}{59} a^{16} + \frac{186}{59} a^{15} - \frac{224}{59} a^{14} + \frac{161}{59} a^{13} - \frac{110}{59} a^{12} + \frac{10}{59} a^{11} + \frac{162}{59} a^{10} - \frac{157}{59} a^{9} + \frac{166}{59} a^{8} - \frac{264}{59} a^{7} + \frac{211}{59} a^{6} - \frac{60}{59} a^{5} + \frac{66}{59} a^{4} - \frac{5}{59} a^{3} - \frac{57}{59} a^{2} + \frac{25}{59} a - \frac{24}{59} \),  \( \frac{29}{59} a^{19} - \frac{46}{59} a^{18} + \frac{70}{59} a^{17} - \frac{143}{59} a^{16} + \frac{183}{59} a^{15} - \frac{228}{59} a^{14} + \frac{245}{59} a^{13} - \frac{211}{59} a^{12} + \frac{164}{59} a^{11} - \frac{128}{59} a^{10} + \frac{33}{59} a^{9} + \frac{32}{59} a^{8} - \frac{58}{59} a^{7} + \frac{62}{59} a^{6} + \frac{19}{59} a^{5} + \frac{44}{59} a^{4} - \frac{23}{59} a^{3} - \frac{38}{59} a^{2} + \frac{56}{59} a - \frac{16}{59} \),  \( \frac{26}{59} a^{19} - \frac{27}{59} a^{18} + \frac{77}{59} a^{17} - \frac{116}{59} a^{16} + \frac{101}{59} a^{15} - \frac{180}{59} a^{14} + \frac{181}{59} a^{13} - \frac{120}{59} a^{12} + \frac{145}{59} a^{11} - \frac{129}{59} a^{10} - \frac{5}{59} a^{9} - \frac{130}{59} a^{8} + \frac{66}{59} a^{7} + \frac{80}{59} a^{6} + \frac{74}{59} a^{5} + \frac{13}{59} a^{4} - \frac{43}{59} a^{3} - \frac{30}{59} a^{2} - \frac{21}{59} a + \frac{65}{59} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 622.109043514 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T1021:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 7257600
The 84 conjugacy class representatives for t20n1021 are not computed
Character table for t20n1021 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 10.0.236438047.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
3Data not computed
236438047Data not computed