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Properties

Label 20.4.10175972532...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 37^{15}$
Root discriminant $50.16$
Ramified primes $5, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5504, 22464, 13408, -209056, 61200, 234968, -433680, 299341, 21067, -111754, 27367, 9955, -2464, 397, -1437, 524, 37, -19, -2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 2*x^18 - 19*x^17 + 37*x^16 + 524*x^15 - 1437*x^14 + 397*x^13 - 2464*x^12 + 9955*x^11 + 27367*x^10 - 111754*x^9 + 21067*x^8 + 299341*x^7 - 433680*x^6 + 234968*x^5 + 61200*x^4 - 209056*x^3 + 13408*x^2 + 22464*x - 5504)
 
gp: K = bnfinit(x^20 - 3*x^19 - 2*x^18 - 19*x^17 + 37*x^16 + 524*x^15 - 1437*x^14 + 397*x^13 - 2464*x^12 + 9955*x^11 + 27367*x^10 - 111754*x^9 + 21067*x^8 + 299341*x^7 - 433680*x^6 + 234968*x^5 + 61200*x^4 - 209056*x^3 + 13408*x^2 + 22464*x - 5504, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 2 x^{18} - 19 x^{17} + 37 x^{16} + 524 x^{15} - 1437 x^{14} + 397 x^{13} - 2464 x^{12} + 9955 x^{11} + 27367 x^{10} - 111754 x^{9} + 21067 x^{8} + 299341 x^{7} - 433680 x^{6} + 234968 x^{5} + 61200 x^{4} - 209056 x^{3} + 13408 x^{2} + 22464 x - 5504 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10175972532709207369643951416015625=5^{15}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.16$
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:  $5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{16} + \frac{1}{24} a^{15} - \frac{1}{12} a^{14} - \frac{7}{24} a^{13} + \frac{1}{24} a^{12} - \frac{1}{2} a^{11} + \frac{7}{24} a^{10} + \frac{3}{8} a^{9} + \frac{7}{24} a^{7} - \frac{5}{24} a^{6} - \frac{5}{12} a^{5} - \frac{1}{24} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{48} a^{17} - \frac{1}{48} a^{16} - \frac{1}{12} a^{15} - \frac{1}{16} a^{14} - \frac{3}{16} a^{13} + \frac{5}{24} a^{12} - \frac{17}{48} a^{11} + \frac{19}{48} a^{10} + \frac{1}{8} a^{9} + \frac{7}{48} a^{8} - \frac{19}{48} a^{7} + \frac{19}{48} a^{5} + \frac{11}{48} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{192} a^{18} + \frac{1}{192} a^{17} - \frac{1}{96} a^{16} + \frac{17}{192} a^{15} + \frac{1}{192} a^{14} + \frac{1}{16} a^{13} - \frac{65}{192} a^{12} - \frac{29}{64} a^{11} - \frac{1}{8} a^{10} - \frac{17}{192} a^{9} + \frac{19}{192} a^{8} + \frac{19}{96} a^{7} - \frac{25}{192} a^{6} - \frac{21}{64} a^{5} + \frac{1}{8} a^{4} - \frac{5}{16} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{26890396929169272335872592635366140987743343356928} a^{19} - \frac{7770323781786371741880327323043776068844215361}{26890396929169272335872592635366140987743343356928} a^{18} + \frac{15676976483196396245353331494369287966308649893}{2240866410764106027989382719613845082311945279744} a^{17} - \frac{546334397903979320789727138117149962350708927259}{26890396929169272335872592635366140987743343356928} a^{16} - \frac{2278012027641186570136191912936874447482292737233}{26890396929169272335872592635366140987743343356928} a^{15} - \frac{1457051569762777709464583451607390223023212544043}{13445198464584636167936296317683070493871671678464} a^{14} + \frac{7780203733933817003683532568112657416227864034743}{26890396929169272335872592635366140987743343356928} a^{13} - \frac{5900978276650722985785955630662059470732040175173}{26890396929169272335872592635366140987743343356928} a^{12} - \frac{4433356698018838538608991195091854658352702906229}{13445198464584636167936296317683070493871671678464} a^{11} - \frac{4443715547214419799100107086657670635378059227889}{26890396929169272335872592635366140987743343356928} a^{10} - \frac{2658179704769994116635549252149489634127152757819}{26890396929169272335872592635366140987743343356928} a^{9} - \frac{163275912538494378370910324242407926738008762121}{420162452018269880248009259927595952933489739952} a^{8} - \frac{11979021200018334963317660540730067571062627596469}{26890396929169272335872592635366140987743343356928} a^{7} - \frac{1051730046317130286055699795401615475143530908509}{26890396929169272335872592635366140987743343356928} a^{6} - \frac{808703480903969861386245492194249162042253844247}{4481732821528212055978765439227690164623890559488} a^{5} + \frac{606445314059140352190538358705467485180108720689}{6722599232292318083968148158841535246935835839232} a^{4} - \frac{419206320039803780918112010780896221557616383933}{3361299616146159041984074079420767623467917919616} a^{3} - \frac{251575851197527035527970039740843851714145833037}{560216602691026506997345679903461270577986319936} a^{2} + \frac{3133202935145217580034210967583375670141053255}{210081226009134940124004629963797976466744869976} a - \frac{184537180681495474586330386939954828662032291573}{420162452018269880248009259927595952933489739952}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 14395657396.556372 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{185}) \), 4.4.6331625.2, 5.1.171125.1, 10.2.5417496640625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $20$

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
37Data not computed