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Properties

Label 20.4.43111000196...4401.1
Degree $20$
Signature $[4, 8]$
Discriminant $401^{11}$
Root discriminant $27.02$
Ramified prime $401$
Class number $1$
Class group Trivial
Galois group $(C_2\times C_2^4:C_5).C_2$ (as 20T84)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 53, -153, -264, 152, 760, 715, -723, -1417, -574, -215, -308, -211, -168, -174, -68, -15, -16, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 7*x^18 - 16*x^17 - 15*x^16 - 68*x^15 - 174*x^14 - 168*x^13 - 211*x^12 - 308*x^11 - 215*x^10 - 574*x^9 - 1417*x^8 - 723*x^7 + 715*x^6 + 760*x^5 + 152*x^4 - 264*x^3 - 153*x^2 + 53*x - 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 7*x^18 - 16*x^17 - 15*x^16 - 68*x^15 - 174*x^14 - 168*x^13 - 211*x^12 - 308*x^11 - 215*x^10 - 574*x^9 - 1417*x^8 - 723*x^7 + 715*x^6 + 760*x^5 + 152*x^4 - 264*x^3 - 153*x^2 + 53*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 7 x^{18} - 16 x^{17} - 15 x^{16} - 68 x^{15} - 174 x^{14} - 168 x^{13} - 211 x^{12} - 308 x^{11} - 215 x^{10} - 574 x^{9} - 1417 x^{8} - 723 x^{7} + 715 x^{6} + 760 x^{5} + 152 x^{4} - 264 x^{3} - 153 x^{2} + 53 x - 1 \)

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:  \(43111000196969091338568804401=401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.02$
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:  $401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{174361938159545922404070025221} a^{19} - \frac{13450379408362264763792249233}{174361938159545922404070025221} a^{18} - \frac{6364897731856316312349303622}{58120646053181974134690008407} a^{17} - \frac{13325794126058276494732521080}{174361938159545922404070025221} a^{16} - \frac{23546576144756480394257916755}{174361938159545922404070025221} a^{15} - \frac{5819007739095945705897113441}{174361938159545922404070025221} a^{14} - \frac{7183423496773617760598829466}{58120646053181974134690008407} a^{13} + \frac{2966559444429930957298157317}{58120646053181974134690008407} a^{12} + \frac{39921817084216743512360237683}{174361938159545922404070025221} a^{11} + \frac{41593298758235594449853124812}{174361938159545922404070025221} a^{10} + \frac{36611772886592358610939848851}{174361938159545922404070025221} a^{9} + \frac{4243394840098515545325853961}{58120646053181974134690008407} a^{8} + \frac{4518738535113291585994744549}{174361938159545922404070025221} a^{7} + \frac{75991412225459240606856478507}{174361938159545922404070025221} a^{6} - \frac{29787779262099738887434757212}{174361938159545922404070025221} a^{5} - \frac{19745049754031889718339669269}{58120646053181974134690008407} a^{4} - \frac{14677397932149076946496854}{876190644017818705548090579} a^{3} - \frac{83162134693840326227809839701}{174361938159545922404070025221} a^{2} - \frac{23578843212007414210659958342}{58120646053181974134690008407} a + \frac{17469075848916210738192934231}{174361938159545922404070025221}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1532201.06636 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_2^4:C_5).C_2$ (as 20T84):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $(C_2\times C_2^4:C_5).C_2$
Character table for $(C_2\times C_2^4:C_5).C_2$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
401Data not computed