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Properties

Label 20.0.12200382642...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{35}\cdot 31^{12}\cdot 181^{12}$
Root discriminant $7150.21$
Ramified primes $3, 5, 31, 181$
Class number Not computed
Class group Not computed
Galois group $C_5\times F_5$ (as 20T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75302901891158409924316939431, 0, 0, 0, 0, -24499004414433966486927, 0, 0, 0, 0, 2737990582811559, 0, 0, 0, 0, -78330153, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 78330153*x^15 + 2737990582811559*x^10 - 24499004414433966486927*x^5 + 75302901891158409924316939431)
 
gp: K = bnfinit(x^20 - 78330153*x^15 + 2737990582811559*x^10 - 24499004414433966486927*x^5 + 75302901891158409924316939431, 1)
 

Normalized defining polynomial

\( x^{20} - 78330153 x^{15} + 2737990582811559 x^{10} - 24499004414433966486927 x^{5} + 75302901891158409924316939431 \)

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:  \(122003826420211264288677099926477115147088065905336526338942348957061767578125=3^{16}\cdot 5^{35}\cdot 31^{12}\cdot 181^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $7150.21$
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:  $3, 5, 31, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{93} a^{8} + \frac{9}{31} a^{3}$, $\frac{1}{93} a^{9} + \frac{9}{31} a^{4}$, $\frac{1}{17220159} a^{10} + \frac{20511}{173941} a^{5} - \frac{3}{11}$, $\frac{1}{17220159} a^{11} + \frac{20511}{173941} a^{6} - \frac{3}{11} a$, $\frac{1}{533824929} a^{12} - \frac{2373641}{16176513} a^{7} + \frac{30}{341} a^{2}$, $\frac{1}{96622312149} a^{13} - \frac{2373641}{2927948853} a^{8} + \frac{19467}{61721} a^{3}$, $\frac{1}{2995291676619} a^{14} - \frac{2373641}{90766414443} a^{9} + \frac{883561}{1913351} a^{4}$, $\frac{1}{1338196231172409953024544392341964943} a^{15} - \frac{12606757049234836739682249614}{446065410390803317674848130780654981} a^{10} - \frac{92614515813996850039302862445}{854821290726555398429827989147} a^{5} + \frac{39861871681246086851}{291952262024897945999}$, $\frac{1}{1338196231172409953024544392341964943} a^{16} - \frac{12606757049234836739682249614}{446065410390803317674848130780654981} a^{11} - \frac{92614515813996850039302862445}{854821290726555398429827989147} a^{6} + \frac{39861871681246086851}{291952262024897945999} a$, $\frac{1}{7508619053108392246420718585430765295173} a^{17} + \frac{557274103435135428880203076484}{2502873017702797415473572861810255098391} a^{12} - \frac{275122054824386914804813737409708}{4796402262266702340589764847103817} a^{7} - \frac{341544284697449350731979}{1638144142221702375000389} a^{2}$, $\frac{1}{232767190646360159639042276148353724150363} a^{18} - \frac{219836160861733115146913277286}{77589063548786719879680758716117908050121} a^{13} - \frac{158471071827768999391864787019238}{148688470130267772558282710260218327} a^{8} - \frac{19118108982337276777357847}{50782468408872773625012059} a^{3}$, $\frac{1}{1306056706716726855734666211468412746207686793} a^{19} + \frac{28688665670981776722661815082958}{435352235572242285244888737156137582069228931} a^{14} + \frac{1914063017267378980783152373099982078}{834291005900932471824524287270085032797} a^{9} - \frac{19075665370938533317586566664}{284940430242185132809942663049} a^{4}$

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

Class group and class number

Not computed

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:  \( \frac{217435832299865}{1338196231172409953024544392341964943} a^{15} - \frac{4713174228091437696949}{446065410390803317674848130780654981} a^{10} + \frac{93138732302166305144585}{284940430242185132809942663049} a^{5} - \frac{228020258318526907615}{291952262024897945999} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times F_5$ (as 20T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 25 conjugacy class representatives for $C_5\times F_5$
Character table for $C_5\times F_5$ is not computed

Intermediate fields

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

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

Sibling fields

Degree 25 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 $20$ R R $20$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{20}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
5Data not computed
$31$31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.4$x^{5} + 10633$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.5$x^{5} - 74431$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
$181$181.5.4.5$x^{5} - 2896$$5$$1$$4$$C_5$$[\ ]_{5}$
181.5.4.3$x^{5} - 724$$5$$1$$4$$C_5$$[\ ]_{5}$
181.5.4.2$x^{5} + 362$$5$$1$$4$$C_5$$[\ ]_{5}$
181.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$