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Properties

Label 20.20.1318976600...4677.1
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 19^{5}\cdot 103^{5}$
Root discriminant $45.29$
Ramified primes $11, 19, 103$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times S_4$ (as 20T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -63, 16, 780, -69, -3988, 445, 9949, -1241, -12749, 1324, 8448, -548, -2859, 80, 473, -4, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 36*x^18 - 4*x^17 + 473*x^16 + 80*x^15 - 2859*x^14 - 548*x^13 + 8448*x^12 + 1324*x^11 - 12749*x^10 - 1241*x^9 + 9949*x^8 + 445*x^7 - 3988*x^6 - 69*x^5 + 780*x^4 + 16*x^3 - 63*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^20 - 36*x^18 - 4*x^17 + 473*x^16 + 80*x^15 - 2859*x^14 - 548*x^13 + 8448*x^12 + 1324*x^11 - 12749*x^10 - 1241*x^9 + 9949*x^8 + 445*x^7 - 3988*x^6 - 69*x^5 + 780*x^4 + 16*x^3 - 63*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 36 x^{18} - 4 x^{17} + 473 x^{16} + 80 x^{15} - 2859 x^{14} - 548 x^{13} + 8448 x^{12} + 1324 x^{11} - 12749 x^{10} - 1241 x^{9} + 9949 x^{8} + 445 x^{7} - 3988 x^{6} - 69 x^{5} + 780 x^{4} + 16 x^{3} - 63 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1318976600112975820532265102854677=11^{16}\cdot 19^{5}\cdot 103^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.29$
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:  $11, 19, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{125459340733043186561} a^{19} - \frac{34993742775803146116}{125459340733043186561} a^{18} + \frac{6541243548098874590}{125459340733043186561} a^{17} + \frac{15296844874956161403}{125459340733043186561} a^{16} + \frac{45019165376074318617}{125459340733043186561} a^{15} + \frac{46196593412902723840}{125459340733043186561} a^{14} - \frac{14708169646595909611}{125459340733043186561} a^{13} - \frac{29343024212548399899}{125459340733043186561} a^{12} - \frac{8530755568975933058}{125459340733043186561} a^{11} - \frac{54508266671024331963}{125459340733043186561} a^{10} - \frac{30596497870558657559}{125459340733043186561} a^{9} - \frac{23164754981856069680}{125459340733043186561} a^{8} - \frac{44190983831015439367}{125459340733043186561} a^{7} - \frac{4734860149840213558}{125459340733043186561} a^{6} - \frac{31879464480684771449}{125459340733043186561} a^{5} + \frac{24388506928855891282}{125459340733043186561} a^{4} + \frac{6646394692860903671}{125459340733043186561} a^{3} + \frac{11309469415681612002}{125459340733043186561} a^{2} - \frac{51814995106137861759}{125459340733043186561} a + \frac{2181459182034255131}{125459340733043186561}$

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:  $19$
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:  \( 13698434875.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_4$ (as 20T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 25 conjugacy class representatives for $C_5\times S_4$
Character table for $C_5\times S_4$ is not computed

Intermediate fields

4.4.1957.1, \(\Q(\zeta_{11})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed
Degree 40 sibling: 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 $20$ $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ $20$ $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ $20$ $15{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ $15{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
11Data not computed
$19$19.10.5.1$x^{10} - 722 x^{6} + 130321 x^{2} - 61902475$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$103$103.10.0.1$x^{10} - x + 12$$1$$10$$0$$C_{10}$$[\ ]^{10}$
103.10.5.2$x^{10} - 112550881 x^{2} + 208669333374$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$