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Properties

Label 20.0.10280287112...8369.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 43^{2}$
Root discriminant $12.61$
Ramified primes $11, 43$
Class number $1$
Class group Trivial
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 26, -65, 121, -188, 271, -367, 450, -495, 507, -495, 450, -367, 271, -188, 121, -65, 26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 65*x^17 + 121*x^16 - 188*x^15 + 271*x^14 - 367*x^13 + 450*x^12 - 495*x^11 + 507*x^10 - 495*x^9 + 450*x^8 - 367*x^7 + 271*x^6 - 188*x^5 + 121*x^4 - 65*x^3 + 26*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 26*x^18 - 65*x^17 + 121*x^16 - 188*x^15 + 271*x^14 - 367*x^13 + 450*x^12 - 495*x^11 + 507*x^10 - 495*x^9 + 450*x^8 - 367*x^7 + 271*x^6 - 188*x^5 + 121*x^4 - 65*x^3 + 26*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 26 x^{18} - 65 x^{17} + 121 x^{16} - 188 x^{15} + 271 x^{14} - 367 x^{13} + 450 x^{12} - 495 x^{11} + 507 x^{10} - 495 x^{9} + 450 x^{8} - 367 x^{7} + 271 x^{6} - 188 x^{5} + 121 x^{4} - 65 x^{3} + 26 x^{2} - 7 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10280287112647136008369=11^{18}\cdot 43^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.61$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{89} a^{18} + \frac{10}{89} a^{17} + \frac{17}{89} a^{16} + \frac{36}{89} a^{15} + \frac{4}{89} a^{14} + \frac{22}{89} a^{13} + \frac{18}{89} a^{12} + \frac{6}{89} a^{11} + \frac{33}{89} a^{9} + \frac{6}{89} a^{7} + \frac{18}{89} a^{6} + \frac{22}{89} a^{5} + \frac{4}{89} a^{4} + \frac{36}{89} a^{3} + \frac{17}{89} a^{2} + \frac{10}{89} a + \frac{1}{89}$, $\frac{1}{5963} a^{19} - \frac{8}{5963} a^{18} + \frac{905}{5963} a^{17} + \frac{1777}{5963} a^{16} + \frac{1225}{5963} a^{15} + \frac{128}{5963} a^{14} - \frac{2336}{5963} a^{13} - \frac{2721}{5963} a^{12} + \frac{960}{5963} a^{11} - \frac{2192}{5963} a^{10} + \frac{2699}{5963} a^{9} + \frac{1697}{5963} a^{8} - \frac{1247}{5963} a^{7} + \frac{143}{5963} a^{6} - \frac{2083}{5963} a^{5} - \frac{2795}{5963} a^{4} + \frac{437}{5963} a^{3} + \frac{1039}{5963} a^{2} + \frac{1868}{5963} a + \frac{872}{5963}$

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

Class group and class number

Trivial group, which has 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:  \( \frac{36609}{5963} a^{19} - \frac{234247}{5963} a^{18} + \frac{807598}{5963} a^{17} - \frac{1873715}{5963} a^{16} + \frac{3235809}{5963} a^{15} - \frac{4787312}{5963} a^{14} + \frac{6796321}{5963} a^{13} - \frac{8999542}{5963} a^{12} + \frac{10582976}{5963} a^{11} - \frac{11141721}{5963} a^{10} + \frac{11154204}{5963} a^{9} - \frac{10694683}{5963} a^{8} + \frac{9339296}{5963} a^{7} - \frac{7162191}{5963} a^{6} + \frac{5056663}{5963} a^{5} - \frac{3429820}{5963} a^{4} + \frac{2086052}{5963} a^{3} - \frac{942576}{5963} a^{2} + \frac{278102}{5963} a - \frac{45600}{5963} \) (order $22$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4651.30276372 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 10.6.101391750713.1, \(\Q(\zeta_{11})\), 10.4.9217431883.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$