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Properties

Label 20.10.9228883441...1648.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 11^{18}\cdot 1583$
Root discriminant $25.02$
Ramified primes $2, 11, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T409

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 17, 42, -310, 748, -819, -180, 2060, -3378, 2804, -718, -1150, 1630, -1032, 308, 38, -84, 41, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 41*x^18 - 84*x^17 + 38*x^16 + 308*x^15 - 1032*x^14 + 1630*x^13 - 1150*x^12 - 718*x^11 + 2804*x^10 - 3378*x^9 + 2060*x^8 - 180*x^7 - 819*x^6 + 748*x^5 - 310*x^4 + 42*x^3 + 17*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 41*x^18 - 84*x^17 + 38*x^16 + 308*x^15 - 1032*x^14 + 1630*x^13 - 1150*x^12 - 718*x^11 + 2804*x^10 - 3378*x^9 + 2060*x^8 - 180*x^7 - 819*x^6 + 748*x^5 - 310*x^4 + 42*x^3 + 17*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 41 x^{18} - 84 x^{17} + 38 x^{16} + 308 x^{15} - 1032 x^{14} + 1630 x^{13} - 1150 x^{12} - 718 x^{11} + 2804 x^{10} - 3378 x^{9} + 2060 x^{8} - 180 x^{7} - 819 x^{6} + 748 x^{5} - 310 x^{4} + 42 x^{3} + 17 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-9228883441492376875877531648=-\,2^{20}\cdot 11^{18}\cdot 1583\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.02$
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:  $2, 11, 1583$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{89} a^{16} - \frac{8}{89} a^{15} - \frac{43}{89} a^{14} - \frac{4}{89} a^{13} - \frac{35}{89} a^{12} - \frac{13}{89} a^{11} + \frac{13}{89} a^{9} + \frac{22}{89} a^{8} - \frac{18}{89} a^{7} + \frac{7}{89} a^{6} + \frac{19}{89} a^{5} + \frac{28}{89} a^{4} + \frac{42}{89} a^{3} - \frac{25}{89} a^{2} + \frac{14}{89} a - \frac{16}{89}$, $\frac{1}{89} a^{17} - \frac{18}{89} a^{15} + \frac{8}{89} a^{14} + \frac{22}{89} a^{13} - \frac{26}{89} a^{12} - \frac{15}{89} a^{11} + \frac{13}{89} a^{10} + \frac{37}{89} a^{9} - \frac{20}{89} a^{8} + \frac{41}{89} a^{7} - \frac{14}{89} a^{6} + \frac{2}{89} a^{5} - \frac{1}{89} a^{4} + \frac{44}{89} a^{3} - \frac{8}{89} a^{2} + \frac{7}{89} a - \frac{39}{89}$, $\frac{1}{3827} a^{18} - \frac{9}{3827} a^{17} + \frac{14}{3827} a^{16} + \frac{92}{3827} a^{15} + \frac{265}{3827} a^{14} - \frac{352}{3827} a^{13} + \frac{167}{3827} a^{12} - \frac{1069}{3827} a^{11} + \frac{365}{3827} a^{10} - \frac{1450}{3827} a^{9} - \frac{1389}{3827} a^{8} - \frac{425}{3827} a^{7} + \frac{708}{3827} a^{6} + \frac{1212}{3827} a^{5} + \frac{1839}{3827} a^{4} - \frac{395}{3827} a^{3} - \frac{9}{3827} a^{2} + \frac{435}{3827} a - \frac{1407}{3827}$, $\frac{1}{3827} a^{19} + \frac{19}{3827} a^{17} + \frac{3}{3827} a^{16} + \frac{1265}{3827} a^{15} + \frac{485}{3827} a^{14} - \frac{249}{3827} a^{13} + \frac{1896}{3827} a^{12} - \frac{97}{3827} a^{11} - \frac{874}{3827} a^{10} + \frac{1256}{3827} a^{9} - \frac{241}{3827} a^{8} + \frac{452}{3827} a^{7} + \frac{1048}{3827} a^{6} + \frac{1180}{3827} a^{5} - \frac{1431}{3827} a^{4} - \frac{1156}{3827} a^{3} + \frac{1214}{3827} a^{2} + \frac{100}{3827} a - \frac{1096}{3827}$

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

Class group and class number

Trivial group, which has 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:  $14$
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:  \( 3068456.42972 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T409:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n409 are not computed
Character table for t20n409 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
11Data not computed
1583Data not computed