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Properties

Label 20.8.25721213848...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{11}\cdot 3469^{5}$
Root discriminant $37.20$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 30, -115, -550, 431, -72, -680, 994, 3425, -5048, 3745, -4938, 4411, -1024, -1166, 1162, -447, 56, 17, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 17*x^18 + 56*x^17 - 447*x^16 + 1162*x^15 - 1166*x^14 - 1024*x^13 + 4411*x^12 - 4938*x^11 + 3745*x^10 - 5048*x^9 + 3425*x^8 + 994*x^7 - 680*x^6 - 72*x^5 + 431*x^4 - 550*x^3 - 115*x^2 + 30*x + 5)
 
gp: K = bnfinit(x^20 - 8*x^19 + 17*x^18 + 56*x^17 - 447*x^16 + 1162*x^15 - 1166*x^14 - 1024*x^13 + 4411*x^12 - 4938*x^11 + 3745*x^10 - 5048*x^9 + 3425*x^8 + 994*x^7 - 680*x^6 - 72*x^5 + 431*x^4 - 550*x^3 - 115*x^2 + 30*x + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 17 x^{18} + 56 x^{17} - 447 x^{16} + 1162 x^{15} - 1166 x^{14} - 1024 x^{13} + 4411 x^{12} - 4938 x^{11} + 3745 x^{10} - 5048 x^{9} + 3425 x^{8} + 994 x^{7} - 680 x^{6} - 72 x^{5} + 431 x^{4} - 550 x^{3} - 115 x^{2} + 30 x + 5 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25721213848857003468800000000000=2^{20}\cdot 5^{11}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.20$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{16} - \frac{1}{5} a^{15} - \frac{1}{5} a^{14} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{1809188860430655620297736275360823695} a^{19} - \frac{137838608948327079233481671449134431}{1809188860430655620297736275360823695} a^{18} + \frac{378194044120760877747019512274489001}{1809188860430655620297736275360823695} a^{17} - \frac{167120503702575904379395763403290702}{1809188860430655620297736275360823695} a^{16} - \frac{93756781769882877882973156306942595}{361837772086131124059547255072164739} a^{15} + \frac{236773191834267728852369202102436086}{1809188860430655620297736275360823695} a^{14} - \frac{165892911719335512932083119809046463}{361837772086131124059547255072164739} a^{13} - \frac{411459830748703825088746120339641619}{1809188860430655620297736275360823695} a^{12} + \frac{292981382295235423470770388140536148}{1809188860430655620297736275360823695} a^{11} + \frac{708581616610343340754220153404726734}{1809188860430655620297736275360823695} a^{10} - \frac{455000929390166369136761398314261668}{1809188860430655620297736275360823695} a^{9} + \frac{197961950041600052828793469237104764}{1809188860430655620297736275360823695} a^{8} + \frac{145492056894698920162857465847111613}{1809188860430655620297736275360823695} a^{7} + \frac{144488391205154575941087795845216979}{1809188860430655620297736275360823695} a^{6} + \frac{98902643030869332343040331105099874}{361837772086131124059547255072164739} a^{5} - \frac{285921397246127818067035291783248486}{1809188860430655620297736275360823695} a^{4} + \frac{132568062829973558012713840182286117}{361837772086131124059547255072164739} a^{3} - \frac{72082984324288301609796088773428976}{361837772086131124059547255072164739} a^{2} - \frac{83165634461323786317486051585954465}{361837772086131124059547255072164739} a + \frac{82554047472021027497566725583623205}{361837772086131124059547255072164739}$

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:  $13$
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:  \( 168055410.969 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed