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Properties

Label 20.12.6752487198...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{58}\cdot 5^{22}\cdot 7^{6}\cdot 17^{4}$
Root discriminant $138.51$
Ramified primes $2, 5, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![576480100, 0, -652111600, 0, -201203800, 0, 199488800, 0, 11872700, 0, -7141820, 0, 66910, 0, 68240, 0, -3590, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 - 3590*x^16 + 68240*x^14 + 66910*x^12 - 7141820*x^10 + 11872700*x^8 + 199488800*x^6 - 201203800*x^4 - 652111600*x^2 + 576480100)
 
gp: K = bnfinit(x^20 + 20*x^18 - 3590*x^16 + 68240*x^14 + 66910*x^12 - 7141820*x^10 + 11872700*x^8 + 199488800*x^6 - 201203800*x^4 - 652111600*x^2 + 576480100, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} - 3590 x^{16} + 68240 x^{14} + 66910 x^{12} - 7141820 x^{10} + 11872700 x^{8} + 199488800 x^{6} - 201203800 x^{4} - 652111600 x^{2} + 576480100 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6752487198279797309440000000000000000000000=2^{58}\cdot 5^{22}\cdot 7^{6}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.51$
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:  $2, 5, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{70} a^{10} + \frac{2}{7} a^{8} - \frac{2}{7} a^{6} - \frac{1}{7} a^{4} - \frac{1}{7} a^{2}$, $\frac{1}{70} a^{11} + \frac{2}{7} a^{9} - \frac{2}{7} a^{7} - \frac{1}{7} a^{5} - \frac{1}{7} a^{3}$, $\frac{1}{490} a^{12} - \frac{1}{490} a^{10} - \frac{9}{49} a^{8} + \frac{6}{49} a^{6} - \frac{1}{49} a^{4} + \frac{2}{7} a^{2}$, $\frac{1}{490} a^{13} - \frac{1}{490} a^{11} - \frac{9}{49} a^{9} + \frac{6}{49} a^{7} - \frac{1}{49} a^{5} + \frac{2}{7} a^{3}$, $\frac{1}{3430} a^{14} - \frac{1}{3430} a^{12} + \frac{4}{1715} a^{10} + \frac{104}{343} a^{8} + \frac{97}{343} a^{6} + \frac{23}{49} a^{4} + \frac{1}{7} a^{2}$, $\frac{1}{3430} a^{15} - \frac{1}{3430} a^{13} + \frac{4}{1715} a^{11} + \frac{104}{343} a^{9} + \frac{97}{343} a^{7} + \frac{23}{49} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{24010} a^{16} - \frac{1}{24010} a^{14} + \frac{4}{12005} a^{12} + \frac{11}{24010} a^{10} - \frac{589}{2401} a^{8} + \frac{121}{343} a^{6} + \frac{22}{49} a^{4} - \frac{1}{7} a^{2}$, $\frac{1}{24010} a^{17} - \frac{1}{24010} a^{15} + \frac{4}{12005} a^{13} + \frac{11}{24010} a^{11} - \frac{589}{2401} a^{9} + \frac{121}{343} a^{7} + \frac{22}{49} a^{5} - \frac{1}{7} a^{3}$, $\frac{1}{16817477553421436557058358778086050} a^{18} - \frac{31750330385708581936295613939}{3363495510684287311411671755617210} a^{16} - \frac{78497829136610155991467336079}{1681747755342143655705835877808605} a^{14} + \frac{900994740240036561197519391873}{1681747755342143655705835877808605} a^{12} + \frac{8775707643103487522618786674538}{1681747755342143655705835877808605} a^{10} + \frac{586613057756683964946791227468}{4903054680297794914594273696235} a^{8} - \frac{1472107645431414467769387236645}{6864276552416912880431983174729} a^{6} - \frac{180308500012316531045241760955}{980610936059558982918854739247} a^{4} - \frac{10176208433132001295904731929}{140087276579936997559836391321} a^{2} - \frac{8718881188534160118251205830}{20012468082848142508548055903}$, $\frac{1}{235444685747900111798817022893204700} a^{19} - \frac{183046678321348392433869392636}{11772234287395005589940851144660235} a^{17} + \frac{762025830343011829367551011847}{23544468574790011179881702289320470} a^{15} + \frac{21633911674070712200053305307381}{23544468574790011179881702289320470} a^{13} + \frac{64485295474344715714479205213978}{11772234287395005589940851144660235} a^{11} + \frac{739486843792429295473476975540977}{3363495510684287311411671755617210} a^{9} - \frac{12193191800146268820028455622790}{48049935866918390163023882223103} a^{7} - \frac{1941307547669611447563316051505}{6864276552416912880431983174729} a^{5} - \frac{355306295666408494547543344267}{980610936059558982918854739247} a^{3} + \frac{15653027488581062449422452988}{140087276579936997559836391321} a$

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:  $15$
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:  \( 163219460304000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.1479680000000000.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$