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Properties

Label 20.0.74651918354...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{15}\cdot 23^{10}$
Root discriminant $27.77$
Ramified primes $3, 5, 23$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3555, 11925, 19110, 20385, 16636, 9188, 4837, 6698, 6413, 1358, -660, 395, 266, -245, 0, 28, -37, -2, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 7*x^18 - 2*x^17 - 37*x^16 + 28*x^15 - 245*x^13 + 266*x^12 + 395*x^11 - 660*x^10 + 1358*x^9 + 6413*x^8 + 6698*x^7 + 4837*x^6 + 9188*x^5 + 16636*x^4 + 20385*x^3 + 19110*x^2 + 11925*x + 3555)
 
gp: K = bnfinit(x^20 - 2*x^19 + 7*x^18 - 2*x^17 - 37*x^16 + 28*x^15 - 245*x^13 + 266*x^12 + 395*x^11 - 660*x^10 + 1358*x^9 + 6413*x^8 + 6698*x^7 + 4837*x^6 + 9188*x^5 + 16636*x^4 + 20385*x^3 + 19110*x^2 + 11925*x + 3555, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 7 x^{18} - 2 x^{17} - 37 x^{16} + 28 x^{15} - 245 x^{13} + 266 x^{12} + 395 x^{11} - 660 x^{10} + 1358 x^{9} + 6413 x^{8} + 6698 x^{7} + 4837 x^{6} + 9188 x^{5} + 16636 x^{4} + 20385 x^{3} + 19110 x^{2} + 11925 x + 3555 \)

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:  \(74651918354942620880126953125=3^{10}\cdot 5^{15}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.77$
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:  $3, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{60} a^{14} - \frac{1}{4} a^{13} - \frac{1}{15} a^{12} + \frac{1}{10} a^{11} + \frac{1}{20} a^{10} - \frac{1}{10} a^{9} - \frac{7}{30} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{29}{60} a^{4} - \frac{1}{4} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{60} a^{15} + \frac{11}{60} a^{13} + \frac{1}{10} a^{12} + \frac{1}{20} a^{11} + \frac{3}{20} a^{10} - \frac{7}{30} a^{9} - \frac{3}{10} a^{7} - \frac{3}{10} a^{6} + \frac{29}{60} a^{5} - \frac{1}{2} a^{4} + \frac{5}{12} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{60} a^{16} - \frac{3}{20} a^{13} - \frac{13}{60} a^{12} + \frac{1}{20} a^{11} + \frac{13}{60} a^{10} + \frac{1}{10} a^{9} + \frac{4}{15} a^{8} - \frac{3}{10} a^{7} + \frac{17}{60} a^{6} + \frac{3}{10} a^{5} + \frac{1}{10} a^{4} - \frac{1}{4} a^{3} - \frac{1}{12} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{300} a^{17} - \frac{1}{300} a^{16} + \frac{1}{300} a^{15} + \frac{1}{150} a^{14} + \frac{11}{150} a^{13} + \frac{19}{150} a^{12} + \frac{19}{300} a^{11} - \frac{1}{12} a^{10} - \frac{2}{15} a^{9} + \frac{71}{150} a^{8} - \frac{103}{300} a^{7} + \frac{23}{60} a^{6} - \frac{61}{300} a^{5} - \frac{31}{150} a^{4} - \frac{2}{15} a^{3} + \frac{1}{3} a^{2} + \frac{1}{4} a - \frac{9}{20}$, $\frac{1}{71100} a^{18} + \frac{59}{71100} a^{17} - \frac{101}{17775} a^{16} + \frac{103}{17775} a^{15} + \frac{148}{17775} a^{14} - \frac{2329}{23700} a^{13} + \frac{4517}{35550} a^{12} + \frac{143}{7110} a^{11} + \frac{23}{2844} a^{10} + \frac{1987}{11850} a^{9} + \frac{31247}{71100} a^{8} + \frac{4739}{14220} a^{7} - \frac{4763}{35550} a^{6} + \frac{17659}{35550} a^{5} - \frac{853}{7110} a^{4} + \frac{157}{948} a^{3} + \frac{98}{237} a^{2} - \frac{17}{395} a + \frac{1}{4}$, $\frac{1}{87626264384938612433617800} a^{19} + \frac{43708462532625088807}{87626264384938612433617800} a^{18} - \frac{10882509830514336207631}{8762626438493861243361780} a^{17} + \frac{64997763392059118489143}{14604377397489768738936300} a^{16} - \frac{30132778984187810838043}{5841750958995907495574520} a^{15} - \frac{22590435374302203280057}{87626264384938612433617800} a^{14} - \frac{15541137101569823341220603}{87626264384938612433617800} a^{13} + \frac{1454097565785598718027977}{21906566096234653108404450} a^{12} + \frac{7116770235239899016081}{4868125799163256246312100} a^{11} + \frac{7150823124297556685463217}{87626264384938612433617800} a^{10} + \frac{12054516460865048199590483}{87626264384938612433617800} a^{9} + \frac{2578037061437488136589589}{17525252876987722486723560} a^{8} + \frac{6739720984454885719636949}{43813132192469306216808900} a^{7} + \frac{958487272612708716984941}{2920875479497953747787260} a^{6} - \frac{5762026061976896317656181}{29208754794979537477872600} a^{5} - \frac{43775398718315017586526299}{87626264384938612433617800} a^{4} - \frac{2785968063358010730298067}{5841750958995907495574520} a^{3} - \frac{284956931116276576189598}{730218869874488436946815} a^{2} - \frac{170567789404241515982897}{973625159832651249262420} a + \frac{4538425463808658773561}{24648738223611424031960}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 1412098.09505 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.595125.1, 5.1.595125.1 x5, 10.2.1770868828125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.595125.1
Degree 10 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$23$23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$