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Properties

Label 20.0.34088221436...3281.2
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 19^{10}$
Root discriminant $37.73$
Ramified primes $11, 19$
Class number $55$ (GRH)
Class group $[55]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9765625, -1953125, -1562500, 703125, 171875, -175000, 625, 34875, -7100, -5555, 2531, -1111, -284, 279, 1, -56, 11, 9, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625)
 
gp: K = bnfinit(x^20 - x^19 - 4*x^18 + 9*x^17 + 11*x^16 - 56*x^15 + x^14 + 279*x^13 - 284*x^12 - 1111*x^11 + 2531*x^10 - 5555*x^9 - 7100*x^8 + 34875*x^7 + 625*x^6 - 175000*x^5 + 171875*x^4 + 703125*x^3 - 1562500*x^2 - 1953125*x + 9765625, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 4 x^{18} + 9 x^{17} + 11 x^{16} - 56 x^{15} + x^{14} + 279 x^{13} - 284 x^{12} - 1111 x^{11} + 2531 x^{10} - 5555 x^{9} - 7100 x^{8} + 34875 x^{7} + 625 x^{6} - 175000 x^{5} + 171875 x^{4} + 703125 x^{3} - 1562500 x^{2} - 1953125 x + 9765625 \)

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:  \(34088221436915805032899514033281=11^{18}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.73$
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:  $11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(209=11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(134,·)$, $\chi_{209}(75,·)$, $\chi_{209}(208,·)$, $\chi_{209}(18,·)$, $\chi_{209}(20,·)$, $\chi_{209}(151,·)$, $\chi_{209}(153,·)$, $\chi_{209}(94,·)$, $\chi_{209}(96,·)$, $\chi_{209}(37,·)$, $\chi_{209}(39,·)$, $\chi_{209}(170,·)$, $\chi_{209}(172,·)$, $\chi_{209}(113,·)$, $\chi_{209}(115,·)$, $\chi_{209}(56,·)$, $\chi_{209}(58,·)$, $\chi_{209}(189,·)$, $\chi_{209}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{12655} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1111}{2531}$, $\frac{1}{63275} a^{12} - \frac{1}{63275} a^{11} - \frac{9}{25} a^{10} - \frac{11}{25} a^{9} + \frac{6}{25} a^{8} - \frac{1}{25} a^{7} - \frac{4}{25} a^{6} + \frac{9}{25} a^{5} + \frac{11}{25} a^{4} - \frac{6}{25} a^{3} + \frac{1}{25} a^{2} - \frac{1111}{12655} a - \frac{284}{2531}$, $\frac{1}{316375} a^{13} - \frac{1}{316375} a^{12} - \frac{4}{316375} a^{11} - \frac{36}{125} a^{10} - \frac{44}{125} a^{9} - \frac{26}{125} a^{8} - \frac{4}{125} a^{7} + \frac{9}{125} a^{6} + \frac{11}{125} a^{5} - \frac{56}{125} a^{4} + \frac{1}{125} a^{3} - \frac{1111}{63275} a^{2} - \frac{284}{12655} a + \frac{279}{2531}$, $\frac{1}{1581875} a^{14} - \frac{1}{1581875} a^{13} - \frac{4}{1581875} a^{12} + \frac{9}{1581875} a^{11} + \frac{81}{625} a^{10} + \frac{99}{625} a^{9} + \frac{121}{625} a^{8} + \frac{9}{625} a^{7} + \frac{11}{625} a^{6} - \frac{56}{625} a^{5} + \frac{1}{625} a^{4} - \frac{1111}{316375} a^{3} - \frac{284}{63275} a^{2} + \frac{279}{12655} a + \frac{1}{2531}$, $\frac{1}{7909375} a^{15} - \frac{1}{7909375} a^{14} - \frac{4}{7909375} a^{13} + \frac{9}{7909375} a^{12} + \frac{11}{7909375} a^{11} - \frac{526}{3125} a^{10} + \frac{121}{3125} a^{9} - \frac{616}{3125} a^{8} + \frac{11}{3125} a^{7} - \frac{56}{3125} a^{6} + \frac{1}{3125} a^{5} - \frac{1111}{1581875} a^{4} - \frac{284}{316375} a^{3} + \frac{279}{63275} a^{2} + \frac{1}{12655} a - \frac{56}{2531}$, $\frac{1}{39546875} a^{16} - \frac{1}{39546875} a^{15} - \frac{4}{39546875} a^{14} + \frac{9}{39546875} a^{13} + \frac{11}{39546875} a^{12} - \frac{56}{39546875} a^{11} + \frac{6371}{15625} a^{10} - \frac{3741}{15625} a^{9} + \frac{3136}{15625} a^{8} - \frac{56}{15625} a^{7} + \frac{1}{15625} a^{6} - \frac{1111}{7909375} a^{5} - \frac{284}{1581875} a^{4} + \frac{279}{316375} a^{3} + \frac{1}{63275} a^{2} - \frac{56}{12655} a + \frac{11}{2531}$, $\frac{1}{197734375} a^{17} - \frac{1}{197734375} a^{16} - \frac{4}{197734375} a^{15} + \frac{9}{197734375} a^{14} + \frac{11}{197734375} a^{13} - \frac{56}{197734375} a^{12} + \frac{1}{197734375} a^{11} + \frac{11884}{78125} a^{10} + \frac{34386}{78125} a^{9} - \frac{15681}{78125} a^{8} + \frac{1}{78125} a^{7} - \frac{1111}{39546875} a^{6} - \frac{284}{7909375} a^{5} + \frac{279}{1581875} a^{4} + \frac{1}{316375} a^{3} - \frac{56}{63275} a^{2} + \frac{11}{12655} a + \frac{9}{2531}$, $\frac{1}{988671875} a^{18} - \frac{1}{988671875} a^{17} - \frac{4}{988671875} a^{16} + \frac{9}{988671875} a^{15} + \frac{11}{988671875} a^{14} - \frac{56}{988671875} a^{13} + \frac{1}{988671875} a^{12} + \frac{279}{988671875} a^{11} + \frac{112511}{390625} a^{10} - \frac{171931}{390625} a^{9} + \frac{1}{390625} a^{8} - \frac{1111}{197734375} a^{7} - \frac{284}{39546875} a^{6} + \frac{279}{7909375} a^{5} + \frac{1}{1581875} a^{4} - \frac{56}{316375} a^{3} + \frac{11}{63275} a^{2} + \frac{9}{12655} a - \frac{4}{2531}$, $\frac{1}{4943359375} a^{19} - \frac{1}{4943359375} a^{18} - \frac{4}{4943359375} a^{17} + \frac{9}{4943359375} a^{16} + \frac{11}{4943359375} a^{15} - \frac{56}{4943359375} a^{14} + \frac{1}{4943359375} a^{13} + \frac{279}{4943359375} a^{12} - \frac{284}{4943359375} a^{11} - \frac{562556}{1953125} a^{10} + \frac{1}{1953125} a^{9} - \frac{1111}{988671875} a^{8} - \frac{284}{197734375} a^{7} + \frac{279}{39546875} a^{6} + \frac{1}{7909375} a^{5} - \frac{56}{1581875} a^{4} + \frac{11}{316375} a^{3} + \frac{9}{63275} a^{2} - \frac{4}{12655} a - \frac{1}{2531}$

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

Class group and class number

$C_{55}$, which has order $55$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{56}{39546875} a^{17} + \frac{308549}{39546875} a^{6} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8845130.03478 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{209}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-11}, \sqrt{-19})\), \(\Q(\zeta_{11})^+\), 10.10.5838511919737409.1, \(\Q(\zeta_{11})\), 10.0.530773810885219.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
19Data not computed