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Properties

Label 20.12.1245907905...1889.1
Degree $20$
Signature $[12, 4]$
Discriminant $17^{2}\cdot 401^{11}$
Root discriminant $35.87$
Ramified primes $17, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 103, 1949, 7241, 5962, -10853, -18920, -1021, 12158, 3101, -823, 527, -1956, -473, 823, -74, -97, 47, -5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 5*x^18 + 47*x^17 - 97*x^16 - 74*x^15 + 823*x^14 - 473*x^13 - 1956*x^12 + 527*x^11 - 823*x^10 + 3101*x^9 + 12158*x^8 - 1021*x^7 - 18920*x^6 - 10853*x^5 + 5962*x^4 + 7241*x^3 + 1949*x^2 + 103*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 - 5*x^18 + 47*x^17 - 97*x^16 - 74*x^15 + 823*x^14 - 473*x^13 - 1956*x^12 + 527*x^11 - 823*x^10 + 3101*x^9 + 12158*x^8 - 1021*x^7 - 18920*x^6 - 10853*x^5 + 5962*x^4 + 7241*x^3 + 1949*x^2 + 103*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 5 x^{18} + 47 x^{17} - 97 x^{16} - 74 x^{15} + 823 x^{14} - 473 x^{13} - 1956 x^{12} + 527 x^{11} - 823 x^{10} + 3101 x^{9} + 12158 x^{8} - 1021 x^{7} - 18920 x^{6} - 10853 x^{5} + 5962 x^{4} + 7241 x^{3} + 1949 x^{2} + 103 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12459079056924067396846384471889=17^{2}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.87$
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:  $17, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{249} a^{18} + \frac{35}{249} a^{17} - \frac{29}{83} a^{16} - \frac{124}{249} a^{15} + \frac{36}{83} a^{14} + \frac{53}{249} a^{13} - \frac{85}{249} a^{12} - \frac{52}{249} a^{11} - \frac{94}{249} a^{10} + \frac{61}{249} a^{9} + \frac{43}{249} a^{8} + \frac{92}{249} a^{7} - \frac{26}{83} a^{6} + \frac{95}{249} a^{5} - \frac{40}{249} a^{4} + \frac{20}{249} a^{3} + \frac{16}{83} a^{2} + \frac{10}{249} a + \frac{10}{83}$, $\frac{1}{675803303450138711121761059393803} a^{19} + \frac{512096590254164934341307784729}{675803303450138711121761059393803} a^{18} + \frac{89753324939339873358453423424318}{675803303450138711121761059393803} a^{17} - \frac{328789244387114294560035315777703}{675803303450138711121761059393803} a^{16} - \frac{320409804995666437958451652471514}{675803303450138711121761059393803} a^{15} - \frac{321316257427545403515537288897613}{675803303450138711121761059393803} a^{14} - \frac{76698798726672178591633690460415}{225267767816712903707253686464601} a^{13} + \frac{30212168798801952092244496523385}{225267767816712903707253686464601} a^{12} + \frac{58046223256339349130042649902937}{225267767816712903707253686464601} a^{11} + \frac{212587730068894210385377622203445}{675803303450138711121761059393803} a^{10} + \frac{65538890081982785854304725752110}{225267767816712903707253686464601} a^{9} - \frac{31823352266514264412337693016356}{675803303450138711121761059393803} a^{8} + \frac{6738942134389989072059041378}{673781957577406491646820597601} a^{7} - \frac{80555555421632654175927757873867}{675803303450138711121761059393803} a^{6} + \frac{65654557962121136214690231246571}{225267767816712903707253686464601} a^{5} + \frac{8612957529965763650856011992832}{225267767816712903707253686464601} a^{4} - \frac{63169284675385721039171267607014}{675803303450138711121761059393803} a^{3} + \frac{303154607284158816094262717006284}{675803303450138711121761059393803} a^{2} - \frac{40400802274658435294216868452290}{675803303450138711121761059393803} a - \frac{40617412680238325531879485037889}{225267767816712903707253686464601}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 252382636.588 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed