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Properties

Label 20.10.2136345869...3423.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{18}\cdot 727^{3}$
Root discriminant $23.25$
Ramified primes $11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T303

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 15, -30, -117, 406, -353, -607, 2337, -3738, 3848, -2641, 920, 382, -961, 947, -637, 332, -137, 42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 42*x^18 - 137*x^17 + 332*x^16 - 637*x^15 + 947*x^14 - 961*x^13 + 382*x^12 + 920*x^11 - 2641*x^10 + 3848*x^9 - 3738*x^8 + 2337*x^7 - 607*x^6 - 353*x^5 + 406*x^4 - 117*x^3 - 30*x^2 + 15*x + 1)
 
gp: K = bnfinit(x^20 - 9*x^19 + 42*x^18 - 137*x^17 + 332*x^16 - 637*x^15 + 947*x^14 - 961*x^13 + 382*x^12 + 920*x^11 - 2641*x^10 + 3848*x^9 - 3738*x^8 + 2337*x^7 - 607*x^6 - 353*x^5 + 406*x^4 - 117*x^3 - 30*x^2 + 15*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 42 x^{18} - 137 x^{17} + 332 x^{16} - 637 x^{15} + 947 x^{14} - 961 x^{13} + 382 x^{12} + 920 x^{11} - 2641 x^{10} + 3848 x^{9} - 3738 x^{8} + 2337 x^{7} - 607 x^{6} - 353 x^{5} + 406 x^{4} - 117 x^{3} - 30 x^{2} + 15 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2136345869968048790230393423=-\,11^{18}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.25$
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:  $11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{465849036778448638789} a^{19} - \frac{229031045066692244648}{465849036778448638789} a^{18} + \frac{54873296091735273172}{465849036778448638789} a^{17} + \frac{60722626166609731178}{465849036778448638789} a^{16} - \frac{142982625187243755065}{465849036778448638789} a^{15} - \frac{197235152693989100245}{465849036778448638789} a^{14} - \frac{29914160663509222425}{465849036778448638789} a^{13} + \frac{17077086019507694656}{465849036778448638789} a^{12} - \frac{136453867858788003311}{465849036778448638789} a^{11} + \frac{1130767284101731506}{465849036778448638789} a^{10} + \frac{186637271437948225037}{465849036778448638789} a^{9} + \frac{91504726029185670444}{465849036778448638789} a^{8} + \frac{50698567625960580796}{465849036778448638789} a^{7} - \frac{135147060178906745073}{465849036778448638789} a^{6} - \frac{18212335544817835666}{465849036778448638789} a^{5} + \frac{73120957522470921199}{465849036778448638789} a^{4} + \frac{229672067522646670531}{465849036778448638789} a^{3} + \frac{185510131849055095306}{465849036778448638789} a^{2} + \frac{137683850848724198558}{465849036778448638789} a - \frac{121065595939071319130}{465849036778448638789}$

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:  $14$
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:  \( 1620893.7055 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T303:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 50 conjugacy class representatives for t20n303 are not computed
Character table for t20n303 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.155838906487.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
11Data not computed
727Data not computed