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Properties

Label 20.20.2470566082...0000.3
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, -229894048766759375, 0, 34983624030843125, 0, -2450150378240625, 0, 86930584311625, 0, -1694385726125, 0, 19269155025, 0, -130885375, 0, 521290, 0, -1120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 1120*x^18 + 521290*x^16 - 130885375*x^14 + 19269155025*x^12 - 1694385726125*x^10 + 86930584311625*x^8 - 2450150378240625*x^6 + 34983624030843125*x^4 - 229894048766759375*x^2 + 525065587402128125)
 
gp: K = bnfinit(x^20 - 1120*x^18 + 521290*x^16 - 130885375*x^14 + 19269155025*x^12 - 1694385726125*x^10 + 86930584311625*x^8 - 2450150378240625*x^6 + 34983624030843125*x^4 - 229894048766759375*x^2 + 525065587402128125, 1)
 

Normalized defining polynomial

\( x^{20} - 1120 x^{18} + 521290 x^{16} - 130885375 x^{14} + 19269155025 x^{12} - 1694385726125 x^{10} + 86930584311625 x^{8} - 2450150378240625 x^{6} + 34983624030843125 x^{4} - 229894048766759375 x^{2} + 525065587402128125 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{333979556798229942612688706777644645011186226999728897125169123494375} a^{18} - \frac{44165676267164835217319448807951143172766799052100425754782105038}{333979556798229942612688706777644645011186226999728897125169123494375} a^{16} + \frac{27983501178206891674987919643580054715700475922127323503207203146}{13359182271929197704507548271105785800447449079989155885006764939775} a^{14} - \frac{104431561567378932470403657367886630879212570463109442030112657741}{66795911359645988522537741355528929002237245399945779425033824698875} a^{12} - \frac{194599551336808774547803410326568819137661148211031591716498863173}{13359182271929197704507548271105785800447449079989155885006764939775} a^{10} + \frac{83610032828270104270432621443378199087065169437683419709788519689}{13359182271929197704507548271105785800447449079989155885006764939775} a^{8} + \frac{261975231491735070923162147107721844242526059722069832883243723692}{2671836454385839540901509654221157160089489815997831177001352987955} a^{6} - \frac{142156012096527983233879829434700977488164234178074344100980794294}{2671836454385839540901509654221157160089489815997831177001352987955} a^{4} - \frac{2831124045289201240225247019341616051863858933980221878804366743}{23233360472920343833926170906270931826865128834763749365229156417} a^{2} - \frac{9607975783686048796400022665235422724782182346432628066433}{41224756555547773783222574685619342446323567585357112828301}$, $\frac{1}{333979556798229942612688706777644645011186226999728897125169123494375} a^{19} - \frac{44165676267164835217319448807951143172766799052100425754782105038}{333979556798229942612688706777644645011186226999728897125169123494375} a^{17} + \frac{27983501178206891674987919643580054715700475922127323503207203146}{13359182271929197704507548271105785800447449079989155885006764939775} a^{15} - \frac{104431561567378932470403657367886630879212570463109442030112657741}{66795911359645988522537741355528929002237245399945779425033824698875} a^{13} - \frac{194599551336808774547803410326568819137661148211031591716498863173}{13359182271929197704507548271105785800447449079989155885006764939775} a^{11} + \frac{83610032828270104270432621443378199087065169437683419709788519689}{13359182271929197704507548271105785800447449079989155885006764939775} a^{9} + \frac{261975231491735070923162147107721844242526059722069832883243723692}{2671836454385839540901509654221157160089489815997831177001352987955} a^{7} - \frac{142156012096527983233879829434700977488164234178074344100980794294}{2671836454385839540901509654221157160089489815997831177001352987955} a^{5} - \frac{2831124045289201240225247019341616051863858933980221878804366743}{23233360472920343833926170906270931826865128834763749365229156417} a^{3} - \frac{9607975783686048796400022665235422724782182346432628066433}{41224756555547773783222574685619342446323567585357112828301} a$

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:  \( 125538763513000000 \) (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 $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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