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Properties

Label 20.20.2950690137...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 7919^{2}$
Root discriminant $74.73$
Ramified primes $2, 5, 11, 7919$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![62710561, 0, -247730077, 0, 327924125, 0, -193568871, 0, 61552481, 0, -11586862, 0, 1345774, 0, -96966, 0, 4203, 0, -100, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 100*x^18 + 4203*x^16 - 96966*x^14 + 1345774*x^12 - 11586862*x^10 + 61552481*x^8 - 193568871*x^6 + 327924125*x^4 - 247730077*x^2 + 62710561)
 
gp: K = bnfinit(x^20 - 100*x^18 + 4203*x^16 - 96966*x^14 + 1345774*x^12 - 11586862*x^10 + 61552481*x^8 - 193568871*x^6 + 327924125*x^4 - 247730077*x^2 + 62710561, 1)
 

Normalized defining polynomial

\( x^{20} - 100 x^{18} + 4203 x^{16} - 96966 x^{14} + 1345774 x^{12} - 11586862 x^{10} + 61552481 x^{8} - 193568871 x^{6} + 327924125 x^{4} - 247730077 x^{2} + 62710561 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(29506901376440972086193367040000000000=2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 7919^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.73$
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:  $2, 5, 11, 7919$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{33} a^{10} + \frac{16}{33} a^{8} + \frac{10}{33} a^{6} - \frac{1}{33} a^{4} + \frac{16}{33} a^{2} + \frac{1}{33}$, $\frac{1}{33} a^{11} + \frac{16}{33} a^{9} + \frac{10}{33} a^{7} - \frac{1}{33} a^{5} + \frac{16}{33} a^{3} + \frac{1}{33} a$, $\frac{1}{33} a^{12} - \frac{5}{11} a^{8} + \frac{4}{33} a^{6} - \frac{1}{33} a^{4} + \frac{3}{11} a^{2} - \frac{16}{33}$, $\frac{1}{33} a^{13} - \frac{5}{11} a^{9} + \frac{4}{33} a^{7} - \frac{1}{33} a^{5} + \frac{3}{11} a^{3} - \frac{16}{33} a$, $\frac{1}{33} a^{14} + \frac{13}{33} a^{8} - \frac{16}{33} a^{6} - \frac{2}{11} a^{4} - \frac{7}{33} a^{2} + \frac{5}{11}$, $\frac{1}{33} a^{15} + \frac{13}{33} a^{9} - \frac{16}{33} a^{7} - \frac{2}{11} a^{5} - \frac{7}{33} a^{3} + \frac{5}{11} a$, $\frac{1}{759} a^{16} + \frac{1}{253} a^{14} + \frac{10}{759} a^{12} - \frac{8}{759} a^{10} + \frac{131}{759} a^{8} - \frac{323}{759} a^{6} - \frac{80}{759} a^{4} - \frac{95}{253} a^{2} - \frac{334}{759}$, $\frac{1}{759} a^{17} + \frac{1}{253} a^{15} + \frac{10}{759} a^{13} - \frac{8}{759} a^{11} + \frac{131}{759} a^{9} - \frac{323}{759} a^{7} - \frac{80}{759} a^{5} - \frac{95}{253} a^{3} - \frac{334}{759} a$, $\frac{1}{740214402329577792161721} a^{18} + \frac{162624369721242391505}{740214402329577792161721} a^{16} + \frac{22242313033185022774}{2925748625808607874157} a^{14} + \frac{568294677063212390341}{67292218393597981105611} a^{12} - \frac{3511162072444241851807}{246738134109859264053907} a^{10} - \frac{87731854647913653193535}{740214402329577792161721} a^{8} + \frac{7914766491275980775267}{32183234883894686615727} a^{6} - \frac{79285942530606547790132}{740214402329577792161721} a^{4} - \frac{224669532779998287413462}{740214402329577792161721} a^{2} - \frac{2311803494501207696}{8497565146306096869}$, $\frac{1}{740214402329577792161721} a^{19} + \frac{162624369721242391505}{740214402329577792161721} a^{17} + \frac{22242313033185022774}{2925748625808607874157} a^{15} + \frac{568294677063212390341}{67292218393597981105611} a^{13} - \frac{3511162072444241851807}{246738134109859264053907} a^{11} - \frac{87731854647913653193535}{740214402329577792161721} a^{9} + \frac{7914766491275980775267}{32183234883894686615727} a^{7} - \frac{79285942530606547790132}{740214402329577792161721} a^{5} - \frac{224669532779998287413462}{740214402329577792161721} a^{3} - \frac{2311803494501207696}{8497565146306096869} a$

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:  $19$
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:  \( 1553660170670 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
7919Data not computed