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Properties

Label 20.20.2470566082...0000.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{15}\cdot 11^{16}\cdot 331^{2}\cdot 39161^{2}$
Root discriminant $234.23$
Ramified primes $2, 5, 11, 331, 39161$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T344

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![525065587402128125, 0, -334901325114803125, 0, 43566322711008125, 0, -2487800964728750, 0, 76277301787625, 0, -1378772819750, 0, 15353076700, 0, -106256150, 0, 444245, 0, -1025, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 1025*x^18 + 444245*x^16 - 106256150*x^14 + 15353076700*x^12 - 1378772819750*x^10 + 76277301787625*x^8 - 2487800964728750*x^6 + 43566322711008125*x^4 - 334901325114803125*x^2 + 525065587402128125)
 
gp: K = bnfinit(x^20 - 1025*x^18 + 444245*x^16 - 106256150*x^14 + 15353076700*x^12 - 1378772819750*x^10 + 76277301787625*x^8 - 2487800964728750*x^6 + 43566322711008125*x^4 - 334901325114803125*x^2 + 525065587402128125, 1)
 

Normalized defining polynomial

\( x^{20} - 1025 x^{18} + 444245 x^{16} - 106256150 x^{14} + 15353076700 x^{12} - 1378772819750 x^{10} + 76277301787625 x^{8} - 2487800964728750 x^{6} + 43566322711008125 x^{4} - 334901325114803125 x^{2} + 525065587402128125 \)

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:  \(247056608274284803549371247668512000000000000000=2^{20}\cdot 5^{15}\cdot 11^{16}\cdot 331^{2}\cdot 39161^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $234.23$
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:  $2, 5, 11, 331, 39161$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{4098989747230530644474611098075822878233921082727636102611146338125} a^{18} + \frac{429818338560486117478692291790393710559011333034744984981567644}{4098989747230530644474611098075822878233921082727636102611146338125} a^{16} + \frac{3047489571419773755099464146792984571994492385521645244044216576}{819797949446106128894922219615164575646784216545527220522229267625} a^{14} + \frac{24713591901788532333654311067717685398895277779982390462690357}{163959589889221225778984443923032915129356843309105444104445853525} a^{12} + \frac{1182531776915065355339504110477923548575046178332137783731311587}{163959589889221225778984443923032915129356843309105444104445853525} a^{10} + \frac{78944259176996531528046408784261474096836472729189595016023454}{163959589889221225778984443923032915129356843309105444104445853525} a^{8} + \frac{1454795030622347111544636121554657233193030593513202303850992501}{32791917977844245155796888784606583025871368661821088820889170705} a^{6} + \frac{2726137351683959315753125594736002636159398773602977995394135727}{32791917977844245155796888784606583025871368661821088820889170705} a^{4} - \frac{1137825418266584292802742181355525675381295579590035843538000462}{6558383595568849031159377756921316605174273732364217764177834141} a^{2} - \frac{151776637246203614133057492468085876460027136209403019456}{505958676253206245034876763445699267604335817824504770351}$, $\frac{1}{4098989747230530644474611098075822878233921082727636102611146338125} a^{19} + \frac{429818338560486117478692291790393710559011333034744984981567644}{4098989747230530644474611098075822878233921082727636102611146338125} a^{17} + \frac{3047489571419773755099464146792984571994492385521645244044216576}{819797949446106128894922219615164575646784216545527220522229267625} a^{15} + \frac{24713591901788532333654311067717685398895277779982390462690357}{163959589889221225778984443923032915129356843309105444104445853525} a^{13} + \frac{1182531776915065355339504110477923548575046178332137783731311587}{163959589889221225778984443923032915129356843309105444104445853525} a^{11} + \frac{78944259176996531528046408784261474096836472729189595016023454}{163959589889221225778984443923032915129356843309105444104445853525} a^{9} + \frac{1454795030622347111544636121554657233193030593513202303850992501}{32791917977844245155796888784606583025871368661821088820889170705} a^{7} + \frac{2726137351683959315753125594736002636159398773602977995394135727}{32791917977844245155796888784606583025871368661821088820889170705} a^{5} - \frac{1137825418266584292802742181355525675381295579590035843538000462}{6558383595568849031159377756921316605174273732364217764177834141} a^{3} - \frac{151776637246203614133057492468085876460027136209403019456}{505958676253206245034876763445699267604335817824504770351} a$

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:  \( 148727338307000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T344:

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 t20n344 are not computed
Character table for t20n344 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 $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$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}$
331Data not computed
39161Data not computed