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Properties

Label 20.8.85263528582...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{8}\cdot 5^{13}\cdot 239^{8}$
Root discriminant $39.49$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T144

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-995, 48025, 93770, -68040, -187816, -47073, 81093, 57138, 5610, -14732, -9262, 681, 1982, 614, -223, -321, 26, 60, -9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 9*x^18 + 60*x^17 + 26*x^16 - 321*x^15 - 223*x^14 + 614*x^13 + 1982*x^12 + 681*x^11 - 9262*x^10 - 14732*x^9 + 5610*x^8 + 57138*x^7 + 81093*x^6 - 47073*x^5 - 187816*x^4 - 68040*x^3 + 93770*x^2 + 48025*x - 995)
 
gp: K = bnfinit(x^20 - 4*x^19 - 9*x^18 + 60*x^17 + 26*x^16 - 321*x^15 - 223*x^14 + 614*x^13 + 1982*x^12 + 681*x^11 - 9262*x^10 - 14732*x^9 + 5610*x^8 + 57138*x^7 + 81093*x^6 - 47073*x^5 - 187816*x^4 - 68040*x^3 + 93770*x^2 + 48025*x - 995, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 9 x^{18} + 60 x^{17} + 26 x^{16} - 321 x^{15} - 223 x^{14} + 614 x^{13} + 1982 x^{12} + 681 x^{11} - 9262 x^{10} - 14732 x^{9} + 5610 x^{8} + 57138 x^{7} + 81093 x^{6} - 47073 x^{5} - 187816 x^{4} - 68040 x^{3} + 93770 x^{2} + 48025 x - 995 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(85263528582144256599415283203125=3^{8}\cdot 5^{13}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.49$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4}$, $\frac{1}{55} a^{17} - \frac{3}{55} a^{16} + \frac{3}{11} a^{15} - \frac{21}{55} a^{14} + \frac{4}{55} a^{13} - \frac{9}{55} a^{12} + \frac{26}{55} a^{11} + \frac{18}{55} a^{10} + \frac{21}{55} a^{9} - \frac{16}{55} a^{8} + \frac{6}{55} a^{7} + \frac{3}{11} a^{6} + \frac{2}{11} a^{5} - \frac{12}{55} a^{4} - \frac{2}{11} a^{3} - \frac{3}{11} a^{2} + \frac{5}{11} a - \frac{3}{11}$, $\frac{1}{55} a^{18} - \frac{1}{11} a^{16} + \frac{24}{55} a^{15} - \frac{4}{55} a^{14} + \frac{14}{55} a^{13} - \frac{12}{55} a^{12} - \frac{3}{55} a^{11} - \frac{13}{55} a^{10} + \frac{5}{11} a^{9} - \frac{9}{55} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{3}{11} a^{5} + \frac{4}{11} a^{4} + \frac{2}{11} a^{3} - \frac{4}{11} a^{2} + \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{10131996223261146770307865485796983816234550355} a^{19} + \frac{5721627603703390413383098928983904126282909}{2026399244652229354061573097159396763246910071} a^{18} + \frac{11690138251459252966801114024163218399990583}{2026399244652229354061573097159396763246910071} a^{17} - \frac{195143309623194061841564685617709259114712566}{10131996223261146770307865485796983816234550355} a^{16} + \frac{3910788054638614081350349225989766662066491746}{10131996223261146770307865485796983816234550355} a^{15} - \frac{2507947734426586291494778682662486077523966}{6099937521529889687120930455025276228919055} a^{14} - \frac{403503725053685604670953236619562052143055727}{921090565751013342755260498708816710566777305} a^{13} + \frac{4143177901389938871690840147316985807543562412}{10131996223261146770307865485796983816234550355} a^{12} - \frac{2628130182468883419235286604696340143561032643}{10131996223261146770307865485796983816234550355} a^{11} + \frac{406550648251946958516829362382958531852595190}{2026399244652229354061573097159396763246910071} a^{10} + \frac{4369274968560535354866579057289735922312017776}{10131996223261146770307865485796983816234550355} a^{9} + \frac{2035810306890368384387136228329431966371537283}{10131996223261146770307865485796983816234550355} a^{8} - \frac{4070957307061764488112877213882614850347307691}{10131996223261146770307865485796983816234550355} a^{7} + \frac{65939982018879204178511757855901524835171695}{2026399244652229354061573097159396763246910071} a^{6} - \frac{432738626043755636436412889913970558335792087}{2026399244652229354061573097159396763246910071} a^{5} + \frac{468330896896656072876989575643892177683456034}{2026399244652229354061573097159396763246910071} a^{4} + \frac{350099371958326820470183801192221983726953720}{2026399244652229354061573097159396763246910071} a^{3} + \frac{848005128424800183537323395614389973027164283}{2026399244652229354061573097159396763246910071} a^{2} + \frac{417973802885730988222603267087466339095950974}{2026399244652229354061573097159396763246910071} a + \frac{913694827880924548371362448384217144555450000}{2026399244652229354061573097159396763246910071}$

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:  $13$
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:  \( 101246878.147 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T144:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n144
Character table for t20n144 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed