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Properties

Label 20.0.11989738398...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 211^{8}$
Root discriminant $28.44$
Ramified primes $5, 211$
Class number $11$
Class group $[11]$
Galois group $C_5\times C_5:C_4$ (as 20T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 52, -206, 643, -1610, 3515, -5906, 7316, -6402, 4112, -2105, 1560, -1564, 1266, -662, 285, -100, 30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 30*x^18 - 100*x^17 + 285*x^16 - 662*x^15 + 1266*x^14 - 1564*x^13 + 1560*x^12 - 2105*x^11 + 4112*x^10 - 6402*x^9 + 7316*x^8 - 5906*x^7 + 3515*x^6 - 1610*x^5 + 643*x^4 - 206*x^3 + 52*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 30*x^18 - 100*x^17 + 285*x^16 - 662*x^15 + 1266*x^14 - 1564*x^13 + 1560*x^12 - 2105*x^11 + 4112*x^10 - 6402*x^9 + 7316*x^8 - 5906*x^7 + 3515*x^6 - 1610*x^5 + 643*x^4 - 206*x^3 + 52*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 30 x^{18} - 100 x^{17} + 285 x^{16} - 662 x^{15} + 1266 x^{14} - 1564 x^{13} + 1560 x^{12} - 2105 x^{11} + 4112 x^{10} - 6402 x^{9} + 7316 x^{8} - 5906 x^{7} + 3515 x^{6} - 1610 x^{5} + 643 x^{4} - 206 x^{3} + 52 x^{2} - 9 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(119897383984074477569580078125=5^{15}\cdot 211^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.44$
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:  $5, 211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{25516101531478115409321351451} a^{19} - \frac{12177086231828137377936589368}{25516101531478115409321351451} a^{18} - \frac{1795886439481918976250142874}{25516101531478115409321351451} a^{17} - \frac{4960720617200039774199711122}{25516101531478115409321351451} a^{16} - \frac{7071753334127443275331373048}{25516101531478115409321351451} a^{15} + \frac{9765403735421310316775727616}{25516101531478115409321351451} a^{14} + \frac{3383712797970198364488693613}{25516101531478115409321351451} a^{13} - \frac{10985577691958565546903529735}{25516101531478115409321351451} a^{12} - \frac{12599525052281458634747210398}{25516101531478115409321351451} a^{11} + \frac{8674999449501835925423432014}{25516101531478115409321351451} a^{10} + \frac{9548837358304190202320704006}{25516101531478115409321351451} a^{9} - \frac{7206491978568027765746015099}{25516101531478115409321351451} a^{8} - \frac{2946610583004055390317011633}{25516101531478115409321351451} a^{7} + \frac{1476860195309978483876733857}{25516101531478115409321351451} a^{6} + \frac{11970822316026096268449393313}{25516101531478115409321351451} a^{5} + \frac{4648168046211173251592696726}{25516101531478115409321351451} a^{4} + \frac{10977686059705535399602748956}{25516101531478115409321351451} a^{3} - \frac{9850140078950667210128931362}{25516101531478115409321351451} a^{2} + \frac{2075118909842730547089526399}{25516101531478115409321351451} a - \frac{1323503824124489186123005508}{25516101531478115409321351451}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{11}$, which has order $11$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{22571724679903564918010168613}{25516101531478115409321351451} a^{19} - \frac{132722803818518972420401894986}{25516101531478115409321351451} a^{18} + \frac{660937943351632777917301132446}{25516101531478115409321351451} a^{17} - \frac{2176146872863253327523718729372}{25516101531478115409321351451} a^{16} + \frac{6163329292154191758054522144711}{25516101531478115409321351451} a^{15} - \frac{14174929960968660456578164693884}{25516101531478115409321351451} a^{14} + \frac{26796452293966799546893305398306}{25516101531478115409321351451} a^{13} - \frac{31906818803566828777577110046632}{25516101531478115409321351451} a^{12} + \frac{31046902150396972333023124704088}{25516101531478115409321351451} a^{11} - \frac{43398464575003232288069918660935}{25516101531478115409321351451} a^{10} + \frac{87256184742595427106872339821160}{25516101531478115409321351451} a^{9} - \frac{133495151015631064907691141219906}{25516101531478115409321351451} a^{8} + \frac{147991147804119656376907617596320}{25516101531478115409321351451} a^{7} - \frac{113874615454068408178373279753586}{25516101531478115409321351451} a^{6} + \frac{63934280704684599669302949635295}{25516101531478115409321351451} a^{5} - \frac{27365860800414060534303526867827}{25516101531478115409321351451} a^{4} + \frac{10512196511645567247864687679765}{25516101531478115409321351451} a^{3} - \frac{3021577645136384335687359283858}{25516101531478115409321351451} a^{2} + \frac{652196580119000971664184978598}{25516101531478115409321351451} a - \frac{72204200234697746687966845982}{25516101531478115409321351451} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 454775.127492 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:C_4$ (as 20T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 40 conjugacy class representatives for $C_5\times C_5:C_4$
Character table for $C_5\times C_5:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.10.6194123253125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
5Data not computed
211Data not computed