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Properties

Label 20.14.1807948901...9712.2
Degree $20$
Signature $[14, 3]$
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![1, 1, -64, 183, 226, -1661, 2314, 499, -4663, 4742, -838, -1809, 1522, -599, 112, 152, -164, 40, 14, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 14*x^18 + 40*x^17 - 164*x^16 + 152*x^15 + 112*x^14 - 599*x^13 + 1522*x^12 - 1809*x^11 - 838*x^10 + 4742*x^9 - 4663*x^8 + 499*x^7 + 2314*x^6 - 1661*x^5 + 226*x^4 + 183*x^3 - 64*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 14*x^18 + 40*x^17 - 164*x^16 + 152*x^15 + 112*x^14 - 599*x^13 + 1522*x^12 - 1809*x^11 - 838*x^10 + 4742*x^9 - 4663*x^8 + 499*x^7 + 2314*x^6 - 1661*x^5 + 226*x^4 + 183*x^3 - 64*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 14 x^{18} + 40 x^{17} - 164 x^{16} + 152 x^{15} + 112 x^{14} - 599 x^{13} + 1522 x^{12} - 1809 x^{11} - 838 x^{10} + 4742 x^{9} - 4663 x^{8} + 499 x^{7} + 2314 x^{6} - 1661 x^{5} + 226 x^{4} + 183 x^{3} - 64 x^{2} + x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{49351034776787016019} a^{19} + \frac{2276220632127094634}{49351034776787016019} a^{18} - \frac{448933508453143674}{49351034776787016019} a^{17} + \frac{1312721888937089258}{49351034776787016019} a^{16} + \frac{10766419560827773672}{49351034776787016019} a^{15} + \frac{14401910828216499413}{49351034776787016019} a^{14} - \frac{3178361345374449101}{49351034776787016019} a^{13} - \frac{19110908520890200579}{49351034776787016019} a^{12} + \frac{1465794383042046602}{49351034776787016019} a^{11} + \frac{16117271260142862654}{49351034776787016019} a^{10} + \frac{6308710755707884893}{49351034776787016019} a^{9} - \frac{19522866349513001578}{49351034776787016019} a^{8} - \frac{24152896198496457994}{49351034776787016019} a^{7} + \frac{2227532635647833207}{49351034776787016019} a^{6} + \frac{1395699365598453225}{49351034776787016019} a^{5} + \frac{8182598021610782899}{49351034776787016019} a^{4} - \frac{17668865766904958412}{49351034776787016019} a^{3} - \frac{6114095956429944693}{49351034776787016019} a^{2} + \frac{4898210404105170669}{49351034776787016019} a + \frac{23420668003039021270}{49351034776787016019}$

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:  $16$
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:  \( 11006269.9622 \) (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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$2$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
11Data not computed
727Data not computed