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Properties

Label 20.0.14022744709...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 11^{16}$
Root discriminant $22.77$
Ramified primes $5, 11$
Class number $5$
Class group $[5]$
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 12, -41, 146, 22, 199, 82, 334, -128, 260, -109, 179, -30, 59, -11, 20, -6, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 6*x^17 + 20*x^16 - 11*x^15 + 59*x^14 - 30*x^13 + 179*x^12 - 109*x^11 + 260*x^10 - 128*x^9 + 334*x^8 + 82*x^7 + 199*x^6 + 22*x^5 + 146*x^4 - 41*x^3 + 12*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 5*x^18 - 6*x^17 + 20*x^16 - 11*x^15 + 59*x^14 - 30*x^13 + 179*x^12 - 109*x^11 + 260*x^10 - 128*x^9 + 334*x^8 + 82*x^7 + 199*x^6 + 22*x^5 + 146*x^4 - 41*x^3 + 12*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 5 x^{18} - 6 x^{17} + 20 x^{16} - 11 x^{15} + 59 x^{14} - 30 x^{13} + 179 x^{12} - 109 x^{11} + 260 x^{10} - 128 x^{9} + 334 x^{8} + 82 x^{7} + 199 x^{6} + 22 x^{5} + 146 x^{4} - 41 x^{3} + 12 x^{2} - 3 x + 1 \)

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:  \(1402274470934209014892578125=5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.77$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(55=5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{55}(1,·)$, $\chi_{55}(3,·)$, $\chi_{55}(4,·)$, $\chi_{55}(9,·)$, $\chi_{55}(12,·)$, $\chi_{55}(14,·)$, $\chi_{55}(16,·)$, $\chi_{55}(23,·)$, $\chi_{55}(26,·)$, $\chi_{55}(27,·)$, $\chi_{55}(31,·)$, $\chi_{55}(34,·)$, $\chi_{55}(36,·)$, $\chi_{55}(37,·)$, $\chi_{55}(38,·)$, $\chi_{55}(42,·)$, $\chi_{55}(47,·)$, $\chi_{55}(48,·)$, $\chi_{55}(49,·)$, $\chi_{55}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{99123971} a^{17} + \frac{48325965}{99123971} a^{16} - \frac{11600658}{99123971} a^{15} + \frac{6656579}{99123971} a^{14} - \frac{7205290}{99123971} a^{13} - \frac{41244584}{99123971} a^{12} - \frac{10414845}{99123971} a^{11} + \frac{10227304}{99123971} a^{10} + \frac{37586797}{99123971} a^{9} + \frac{3892955}{99123971} a^{8} - \frac{39557243}{99123971} a^{7} + \frac{25535606}{99123971} a^{6} + \frac{34693279}{99123971} a^{5} - \frac{1364275}{99123971} a^{4} + \frac{34151070}{99123971} a^{3} + \frac{32353695}{99123971} a^{2} - \frac{7897282}{99123971} a - \frac{35616978}{99123971}$, $\frac{1}{99123971} a^{18} + \frac{14706952}{99123971} a^{16} + \frac{34456124}{99123971} a^{15} + \frac{24371681}{99123971} a^{14} - \frac{35094563}{99123971} a^{13} - \frac{839430}{99123971} a^{12} + \frac{30606824}{99123971} a^{11} + \frac{14092870}{99123971} a^{10} - \frac{11519798}{99123971} a^{9} + \frac{7266067}{99123971} a^{8} + \frac{5265686}{99123971} a^{7} + \frac{7963237}{99123971} a^{6} + \frac{27655504}{99123971} a^{5} + \frac{9716099}{99123971} a^{4} + \frac{7540135}{99123971} a^{3} - \frac{9056992}{99123971} a^{2} - \frac{5019034}{99123971} a + \frac{19540355}{99123971}$, $\frac{1}{99123971} a^{19} - \frac{18173108}{99123971} a^{16} + \frac{49216346}{99123971} a^{15} - \frac{17205099}{99123971} a^{14} - \frac{33117074}{99123971} a^{13} + \frac{45022233}{99123971} a^{12} + \frac{17275357}{99123971} a^{11} + \frac{17289672}{99123971} a^{10} + \frac{14252269}{99123971} a^{9} + \frac{12972271}{99123971} a^{8} - \frac{33700107}{99123971} a^{7} - \frac{14273679}{99123971} a^{6} + \frac{34173398}{99123971} a^{5} - \frac{42357972}{99123971} a^{4} + \frac{31042238}{99123971} a^{3} - \frac{41159200}{99123971} a^{2} + \frac{22288525}{99123971} a + \frac{30420541}{99123971}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( -\frac{27559952}{99123971} a^{19} + \frac{27731652}{99123971} a^{18} - \frac{138658260}{99123971} a^{17} + \frac{166389912}{99123971} a^{16} - \frac{554633040}{99123971} a^{15} + \frac{307903757}{99123971} a^{14} - \frac{1636167468}{99123971} a^{13} + \frac{831949560}{99123971} a^{12} - \frac{4963965708}{99123971} a^{11} + \frac{3022750068}{99123971} a^{10} - \frac{7261871926}{99123971} a^{9} + \frac{3549651456}{99123971} a^{8} - \frac{9262371768}{99123971} a^{7} - \frac{2273995464}{99123971} a^{6} - \frac{5518598748}{99123971} a^{5} - \frac{868193439}{99123971} a^{4} - \frac{4048821192}{99123971} a^{3} + \frac{1136997732}{99123971} a^{2} - \frac{332779824}{99123971} a + \frac{83194956}{99123971} \) (order $10$)
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:  \( 140644.599182 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$