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Properties

Label 20.0.17780219642...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 17^{12}$
Root discriminant $18.30$
Ramified primes $5, 17$
Class number $1$
Class group Trivial
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -2, -19, -73, 70, 620, -1178, 143, 1384, -1228, -135, 780, -414, -39, 153, -80, 5, 15, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 15*x^18 + 5*x^17 - 80*x^16 + 153*x^15 - 39*x^14 - 414*x^13 + 780*x^12 - 135*x^11 - 1228*x^10 + 1384*x^9 + 143*x^8 - 1178*x^7 + 620*x^6 + 70*x^5 - 73*x^4 - 19*x^3 - 2*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 15*x^18 + 5*x^17 - 80*x^16 + 153*x^15 - 39*x^14 - 414*x^13 + 780*x^12 - 135*x^11 - 1228*x^10 + 1384*x^9 + 143*x^8 - 1178*x^7 + 620*x^6 + 70*x^5 - 73*x^4 - 19*x^3 - 2*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 15 x^{18} + 5 x^{17} - 80 x^{16} + 153 x^{15} - 39 x^{14} - 414 x^{13} + 780 x^{12} - 135 x^{11} - 1228 x^{10} + 1384 x^{9} + 143 x^{8} - 1178 x^{7} + 620 x^{6} + 70 x^{5} - 73 x^{4} - 19 x^{3} - 2 x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17780219642021514892578125=5^{15}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.30$
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:  $5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{18} + \frac{1}{10} a^{17} + \frac{1}{5} a^{16} - \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{10} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{10} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{3}{10} a^{2} + \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{1839637222550} a^{19} - \frac{85281536033}{1839637222550} a^{18} + \frac{444039828143}{1839637222550} a^{17} - \frac{78549601509}{919818611275} a^{16} - \frac{388001910347}{1839637222550} a^{15} - \frac{4856449643}{73585488902} a^{14} + \frac{159678567173}{919818611275} a^{13} + \frac{20694200553}{183963722255} a^{12} + \frac{83948056181}{367927444510} a^{11} - \frac{23417085557}{183963722255} a^{10} + \frac{365210027536}{919818611275} a^{9} + \frac{723564700807}{1839637222550} a^{8} + \frac{244839281528}{919818611275} a^{7} - \frac{361088779732}{919818611275} a^{6} - \frac{29380060091}{1839637222550} a^{5} - \frac{23359070069}{1839637222550} a^{4} - \frac{294451483219}{1839637222550} a^{3} + \frac{7341750226}{183963722255} a^{2} + \frac{421039299124}{919818611275} a + \frac{761147607821}{1839637222550}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1114810435414}{919818611275} a^{19} - \frac{15248690824899}{1839637222550} a^{18} + \frac{32650210486729}{1839637222550} a^{17} + \frac{3311056976773}{919818611275} a^{16} - \frac{81358163062233}{919818611275} a^{15} + \frac{6724829457233}{36792744451} a^{14} - \frac{68483221780881}{919818611275} a^{13} - \frac{160080799070407}{367927444510} a^{12} + \frac{167178623428114}{183963722255} a^{11} - \frac{60060416875441}{183963722255} a^{10} - \frac{2156377529836559}{1839637222550} a^{9} + \frac{1541203952074473}{919818611275} a^{8} - \frac{301192507647266}{919818611275} a^{7} - \frac{1912973610766267}{1839637222550} a^{6} + \frac{869227104905101}{919818611275} a^{5} - \frac{234968235852591}{919818611275} a^{4} - \frac{9591839450691}{919818611275} a^{3} - \frac{1859344409529}{367927444510} a^{2} + \frac{2797540294747}{919818611275} a + \frac{7056452069613}{1839637222550} \) (order $10$)
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:  \( 171082.874915 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.614125.1, 10.2.1885747578125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ $20$ R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
17Data not computed