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Properties

Label 20.8.34918889042...5625.2
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 11^{10}\cdot 13^{10}$
Root discriminant $26.74$
Ramified primes $5, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -13, 57, -69, -264, 1324, -2598, 1974, 1589, -4574, 3004, 696, -1641, 305, 336, -83, -83, 41, 3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 3*x^18 + 41*x^17 - 83*x^16 - 83*x^15 + 336*x^14 + 305*x^13 - 1641*x^12 + 696*x^11 + 3004*x^10 - 4574*x^9 + 1589*x^8 + 1974*x^7 - 2598*x^6 + 1324*x^5 - 264*x^4 - 69*x^3 + 57*x^2 - 13*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 3*x^18 + 41*x^17 - 83*x^16 - 83*x^15 + 336*x^14 + 305*x^13 - 1641*x^12 + 696*x^11 + 3004*x^10 - 4574*x^9 + 1589*x^8 + 1974*x^7 - 2598*x^6 + 1324*x^5 - 264*x^4 - 69*x^3 + 57*x^2 - 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 3 x^{18} + 41 x^{17} - 83 x^{16} - 83 x^{15} + 336 x^{14} + 305 x^{13} - 1641 x^{12} + 696 x^{11} + 3004 x^{10} - 4574 x^{9} + 1589 x^{8} + 1974 x^{7} - 2598 x^{6} + 1324 x^{5} - 264 x^{4} - 69 x^{3} + 57 x^{2} - 13 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(34918889042392681418447265625=5^{10}\cdot 11^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.74$
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:  $5, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{571} a^{18} + \frac{91}{571} a^{17} - \frac{220}{571} a^{16} + \frac{167}{571} a^{15} - \frac{151}{571} a^{14} + \frac{134}{571} a^{13} + \frac{177}{571} a^{12} - \frac{97}{571} a^{11} - \frac{180}{571} a^{10} - \frac{64}{571} a^{9} - \frac{169}{571} a^{8} - \frac{150}{571} a^{7} + \frac{101}{571} a^{6} - \frac{34}{571} a^{5} + \frac{24}{571} a^{4} - \frac{106}{571} a^{3} + \frac{89}{571} a^{2} - \frac{9}{571} a + \frac{100}{571}$, $\frac{1}{1345007991421957976791} a^{19} + \frac{750875743056355623}{1345007991421957976791} a^{18} + \frac{215022178470016913645}{1345007991421957976791} a^{17} - \frac{208104392384470520811}{1345007991421957976791} a^{16} - \frac{533914925737867935286}{1345007991421957976791} a^{15} + \frac{59415453607423621110}{1345007991421957976791} a^{14} - \frac{497958483765425910900}{1345007991421957976791} a^{13} - \frac{518029689673025725682}{1345007991421957976791} a^{12} - \frac{548042498308268357752}{1345007991421957976791} a^{11} + \frac{319085431227224486359}{1345007991421957976791} a^{10} - \frac{366091121364111508934}{1345007991421957976791} a^{9} - \frac{125687046894165762081}{1345007991421957976791} a^{8} + \frac{281019792027434765727}{1345007991421957976791} a^{7} - \frac{632836302029477162077}{1345007991421957976791} a^{6} - \frac{388935178535222565262}{1345007991421957976791} a^{5} + \frac{364833875715803304596}{1345007991421957976791} a^{4} - \frac{372798668272598099058}{1345007991421957976791} a^{3} + \frac{154008036456469751720}{1345007991421957976791} a^{2} + \frac{224262866752164997043}{1345007991421957976791} a - \frac{632297992251172341577}{1345007991421957976791}$

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:  $13$
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:  \( 4235047.62992 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.1306755003125.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.7.2$x^{8} - 11$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$