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Properties

Label 20.4.15418170861...0000.6
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{16}$
Root discriminant $28.80$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![593, -922, -5168, 16048, -19333, 18682, -16072, 11000, -12376, 9012, -5808, 2394, -119, -304, 220, -112, 75, -44, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 18*x^18 - 44*x^17 + 75*x^16 - 112*x^15 + 220*x^14 - 304*x^13 - 119*x^12 + 2394*x^11 - 5808*x^10 + 9012*x^9 - 12376*x^8 + 11000*x^7 - 16072*x^6 + 18682*x^5 - 19333*x^4 + 16048*x^3 - 5168*x^2 - 922*x + 593)
 
gp: K = bnfinit(x^20 - 6*x^19 + 18*x^18 - 44*x^17 + 75*x^16 - 112*x^15 + 220*x^14 - 304*x^13 - 119*x^12 + 2394*x^11 - 5808*x^10 + 9012*x^9 - 12376*x^8 + 11000*x^7 - 16072*x^6 + 18682*x^5 - 19333*x^4 + 16048*x^3 - 5168*x^2 - 922*x + 593, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 18 x^{18} - 44 x^{17} + 75 x^{16} - 112 x^{15} + 220 x^{14} - 304 x^{13} - 119 x^{12} + 2394 x^{11} - 5808 x^{10} + 9012 x^{9} - 12376 x^{8} + 11000 x^{7} - 16072 x^{6} + 18682 x^{5} - 19333 x^{4} + 16048 x^{3} - 5168 x^{2} - 922 x + 593 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.80$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3391314241051568933966668795349337128249} a^{19} - \frac{125121593770501294804242026601502622535}{3391314241051568933966668795349337128249} a^{18} - \frac{75728729346052511704900664691118361583}{3391314241051568933966668795349337128249} a^{17} + \frac{1580981457020432232746553716547435387042}{3391314241051568933966668795349337128249} a^{16} - \frac{1482342390016913624177082251927713718269}{3391314241051568933966668795349337128249} a^{15} + \frac{1673823162503357971953747485915462773832}{3391314241051568933966668795349337128249} a^{14} + \frac{341062193085952486838003207000546753967}{3391314241051568933966668795349337128249} a^{13} - \frac{377370568420514920179625246050163758927}{3391314241051568933966668795349337128249} a^{12} - \frac{988224708257869920744813203763552604520}{3391314241051568933966668795349337128249} a^{11} + \frac{1282131523854432204754481905076681040725}{3391314241051568933966668795349337128249} a^{10} - \frac{449143663274224370856328356313601254374}{3391314241051568933966668795349337128249} a^{9} + \frac{1523197317841369016264963733468959447470}{3391314241051568933966668795349337128249} a^{8} - \frac{1657898478924858429335274040912548247504}{3391314241051568933966668795349337128249} a^{7} - \frac{30379281346289072781957663408184959410}{3391314241051568933966668795349337128249} a^{6} + \frac{1130300612501273554929538727079987115264}{3391314241051568933966668795349337128249} a^{5} + \frac{765573687777513417588685065545445320}{38104654393837853190636728037633001441} a^{4} - \frac{1504933159847295263700091776160906352252}{3391314241051568933966668795349337128249} a^{3} - \frac{1492488391021823948404631637220210751807}{3391314241051568933966668795349337128249} a^{2} - \frac{1374582956547622208642577381304722152214}{3391314241051568933966668795349337128249} a + \frac{376292010819993694483371255426848491872}{3391314241051568933966668795349337128249}$

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:  $11$
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:  \( 4631387.3319 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

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 t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.219503494144.1

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 $20$ R $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$