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Properties

Label 20.10.1807948901...9712.5
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 727^{3}$
Root discriminant $25.87$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, -456, 271, 4257, 79, -11252, 3162, 13211, -9004, -6050, 8065, -376, -3006, 1230, 370, -375, 42, 43, -13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 13*x^18 + 43*x^17 + 42*x^16 - 375*x^15 + 370*x^14 + 1230*x^13 - 3006*x^12 - 376*x^11 + 8065*x^10 - 6050*x^9 - 9004*x^8 + 13211*x^7 + 3162*x^6 - 11252*x^5 + 79*x^4 + 4257*x^3 + 271*x^2 - 456*x - 67)
 
gp: K = bnfinit(x^20 - x^19 - 13*x^18 + 43*x^17 + 42*x^16 - 375*x^15 + 370*x^14 + 1230*x^13 - 3006*x^12 - 376*x^11 + 8065*x^10 - 6050*x^9 - 9004*x^8 + 13211*x^7 + 3162*x^6 - 11252*x^5 + 79*x^4 + 4257*x^3 + 271*x^2 - 456*x - 67, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 13 x^{18} + 43 x^{17} + 42 x^{16} - 375 x^{15} + 370 x^{14} + 1230 x^{13} - 3006 x^{12} - 376 x^{11} + 8065 x^{10} - 6050 x^{9} - 9004 x^{8} + 13211 x^{7} + 3162 x^{6} - 11252 x^{5} + 79 x^{4} + 4257 x^{3} + 271 x^{2} - 456 x - 67 \)

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:  \(-18079489015266793067734899712=-\,2^{10}\cdot 11^{16}\cdot 727^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{15618280623171620130557091847619} a^{19} - \frac{3244930949674940941505615342289}{15618280623171620130557091847619} a^{18} + \frac{835409944570474613433624674519}{15618280623171620130557091847619} a^{17} - \frac{6180446787981738975465280859889}{15618280623171620130557091847619} a^{16} - \frac{3073181857671879080883911589802}{15618280623171620130557091847619} a^{15} - \frac{5196105001186028670658920385722}{15618280623171620130557091847619} a^{14} + \frac{4602316194999206349784024764380}{15618280623171620130557091847619} a^{13} + \frac{880004024523020425795777046154}{15618280623171620130557091847619} a^{12} + \frac{7232986731776206568703985427087}{15618280623171620130557091847619} a^{11} - \frac{6613188462213950933681825866879}{15618280623171620130557091847619} a^{10} + \frac{6091233029902766067871481653046}{15618280623171620130557091847619} a^{9} - \frac{6164417196810795628470769794664}{15618280623171620130557091847619} a^{8} + \frac{3476856886649604368812950478661}{15618280623171620130557091847619} a^{7} + \frac{4682637900744795962554148220064}{15618280623171620130557091847619} a^{6} + \frac{3793447445370485603373255403109}{15618280623171620130557091847619} a^{5} + \frac{2625497105087782926832748950671}{15618280623171620130557091847619} a^{4} + \frac{165982323830091389003127817644}{15618280623171620130557091847619} a^{3} - \frac{1508818684735340212181685691507}{15618280623171620130557091847619} a^{2} + \frac{5377648090102135694686108683661}{15618280623171620130557091847619} a - \frac{1311469621327513325254448682302}{15618280623171620130557091847619}$

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:  \( 4528638.55855 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
11Data not computed
727Data not computed