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Properties

Label 20.4.12357827945...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 19^{7}\cdot 1699^{5}$
Root discriminant $40.23$
Ramified primes $5, 19, 1699$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-289, 1802, -8878, 30056, -59393, 104994, -138321, 159241, -136668, 92625, -49524, 18232, -4578, -141, 714, -491, 237, -91, 29, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 29*x^18 - 91*x^17 + 237*x^16 - 491*x^15 + 714*x^14 - 141*x^13 - 4578*x^12 + 18232*x^11 - 49524*x^10 + 92625*x^9 - 136668*x^8 + 159241*x^7 - 138321*x^6 + 104994*x^5 - 59393*x^4 + 30056*x^3 - 8878*x^2 + 1802*x - 289)
 
gp: K = bnfinit(x^20 - 6*x^19 + 29*x^18 - 91*x^17 + 237*x^16 - 491*x^15 + 714*x^14 - 141*x^13 - 4578*x^12 + 18232*x^11 - 49524*x^10 + 92625*x^9 - 136668*x^8 + 159241*x^7 - 138321*x^6 + 104994*x^5 - 59393*x^4 + 30056*x^3 - 8878*x^2 + 1802*x - 289, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 29 x^{18} - 91 x^{17} + 237 x^{16} - 491 x^{15} + 714 x^{14} - 141 x^{13} - 4578 x^{12} + 18232 x^{11} - 49524 x^{10} + 92625 x^{9} - 136668 x^{8} + 159241 x^{7} - 138321 x^{6} + 104994 x^{5} - 59393 x^{4} + 30056 x^{3} - 8878 x^{2} + 1802 x - 289 \)

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:  \(123578279458495716101657822265625=5^{10}\cdot 19^{7}\cdot 1699^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.23$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13619554370241990705905109973308697061346274537} a^{19} - \frac{465458458817875609001105822398425631395303743}{1047658028480153131223469997946822850872790349} a^{18} - \frac{1362455186005239029555561531652098750482904823}{13619554370241990705905109973308697061346274537} a^{17} - \frac{4345882724331032902707315033087810499590708885}{13619554370241990705905109973308697061346274537} a^{16} + \frac{149038401891370248126868837483109232547122740}{13619554370241990705905109973308697061346274537} a^{15} - \frac{6775375838251434591284704321331207114226743557}{13619554370241990705905109973308697061346274537} a^{14} + \frac{297079006196312463726107340462109655282079332}{801150257073058276817947645488746885961545561} a^{13} - \frac{6259982068463234962157230978680264765618965820}{13619554370241990705905109973308697061346274537} a^{12} + \frac{5519863502523574558400666993761471156391406214}{13619554370241990705905109973308697061346274537} a^{11} - \frac{1332092664819752797059276651830228689574437972}{13619554370241990705905109973308697061346274537} a^{10} - \frac{399327976000195658609374428123379117489707506}{13619554370241990705905109973308697061346274537} a^{9} - \frac{4263263326600241134636466525420258528177495290}{13619554370241990705905109973308697061346274537} a^{8} - \frac{4865827591293032953628710381272206790789467699}{13619554370241990705905109973308697061346274537} a^{7} - \frac{371527673210563146963790361920712198634677867}{1047658028480153131223469997946822850872790349} a^{6} - \frac{4212885379617843504476605810611999874400025072}{13619554370241990705905109973308697061346274537} a^{5} - \frac{330556670745270025929649334973648919807859498}{1047658028480153131223469997946822850872790349} a^{4} + \frac{3511408504462589583352946638991752270443332119}{13619554370241990705905109973308697061346274537} a^{3} - \frac{244934704131427409658771300097778420294812787}{801150257073058276817947645488746885961545561} a^{2} + \frac{3861068973471284144683735142717472886013286521}{13619554370241990705905109973308697061346274537} a - \frac{38256842995180866129149233169942742379735501}{801150257073058276817947645488746885961545561}$

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:  $11$
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:  \( 26358711.7591 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
1699Data not computed