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Properties

Label 20.0.239...125.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.393\times 10^{21}$
Root discriminant $11.72$
Ramified primes $5, 280001$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 20T654

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 + 9*x^18 - 18*x^17 + 34*x^16 - 53*x^15 + 77*x^14 - 97*x^13 + 114*x^12 - 121*x^11 + 116*x^10 - 104*x^9 + 79*x^8 - 56*x^7 + 33*x^6 - 19*x^5 + 12*x^4 - 7*x^3 + 5*x^2 - 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 5, -7, 12, -19, 33, -56, 79, -104, 116, -121, 114, -97, 77, -53, 34, -18, 9, -3, 1]);
 

\(x^{20} - 3 x^{19} + 9 x^{18} - 18 x^{17} + 34 x^{16} - 53 x^{15} + 77 x^{14} - 97 x^{13} + 114 x^{12} - 121 x^{11} + 116 x^{10} - 104 x^{9} + 79 x^{8} - 56 x^{7} + 33 x^{6} - 19 x^{5} + 12 x^{4} - 7 x^{3} + 5 x^{2} - 2 x + 1\)  Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2392595214874267578125\)\(\medspace = 5^{15}\cdot 280001^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.72$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 280001$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{17} - \frac{1}{7} a^{15} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{45787} a^{19} + \frac{1453}{45787} a^{18} - \frac{3707}{45787} a^{17} + \frac{387}{1477} a^{16} + \frac{22819}{45787} a^{15} - \frac{3869}{45787} a^{14} - \frac{198}{6541} a^{13} + \frac{16138}{45787} a^{12} - \frac{1616}{6541} a^{11} - \frac{216}{1477} a^{10} - \frac{22793}{45787} a^{9} - \frac{4219}{45787} a^{8} + \frac{2691}{6541} a^{7} - \frac{6338}{45787} a^{6} - \frac{18367}{45787} a^{5} - \frac{15845}{45787} a^{4} + \frac{6340}{45787} a^{3} - \frac{11400}{45787} a^{2} - \frac{554}{6541} a + \frac{18136}{45787}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{19365}{45787} a^{19} + \frac{80529}{45787} a^{18} - \frac{197650}{45787} a^{17} + \frac{13336}{1477} a^{16} - \frac{699485}{45787} a^{15} + \frac{1068754}{45787} a^{14} - \frac{1430312}{45787} a^{13} + \frac{242521}{6541} a^{12} - \frac{1799985}{45787} a^{11} + \frac{56946}{1477} a^{10} - \frac{1543911}{45787} a^{9} + \frac{1089651}{45787} a^{8} - \frac{765314}{45787} a^{7} + \frac{248604}{45787} a^{6} - \frac{74953}{45787} a^{5} + \frac{65525}{45787} a^{4} - \frac{25694}{45787} a^{3} + \frac{93824}{45787} a^{2} + \frac{19872}{45787} a + \frac{15355}{45787} \) (order $10$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 858.376938924 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 858.376938924 \cdot 1}{10\sqrt{2392595214874267578125}}\approx 0.168283722445$ (assuming GRH)

Galois group

20T654:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n654 are not computed
Character table for t20n654 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.875003125.1

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 24 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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
280001Data not computed