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Properties

Label 20.0.33630250781...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{16}\cdot 5^{23}$
Root discriminant $26.69$
Ramified primes $2, 3, 5$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, -15, 115, -20, -841, 1115, 1785, -3880, -1055, 5601, -1055, -3880, 1785, 1115, -841, -20, 115, -15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 15*x^18 + 115*x^17 - 20*x^16 - 841*x^15 + 1115*x^14 + 1785*x^13 - 3880*x^12 - 1055*x^11 + 5601*x^10 - 1055*x^9 - 3880*x^8 + 1785*x^7 + 1115*x^6 - 841*x^5 - 20*x^4 + 115*x^3 - 15*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 - 15*x^18 + 115*x^17 - 20*x^16 - 841*x^15 + 1115*x^14 + 1785*x^13 - 3880*x^12 - 1055*x^11 + 5601*x^10 - 1055*x^9 - 3880*x^8 + 1785*x^7 + 1115*x^6 - 841*x^5 - 20*x^4 + 115*x^3 - 15*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 15 x^{18} + 115 x^{17} - 20 x^{16} - 841 x^{15} + 1115 x^{14} + 1785 x^{13} - 3880 x^{12} - 1055 x^{11} + 5601 x^{10} - 1055 x^{9} - 3880 x^{8} + 1785 x^{7} + 1115 x^{6} - 841 x^{5} - 20 x^{4} + 115 x^{3} - 15 x^{2} - 5 x + 1 \)

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:  \(33630250781250000000000000000=2^{16}\cdot 3^{16}\cdot 5^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.69$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{192601} a^{18} + \frac{4397}{192601} a^{17} + \frac{95478}{192601} a^{16} + \frac{34492}{192601} a^{15} - \frac{31302}{192601} a^{14} + \frac{75579}{192601} a^{13} - \frac{83353}{192601} a^{12} - \frac{88795}{192601} a^{11} - \frac{8688}{192601} a^{10} - \frac{21838}{192601} a^{9} - \frac{8688}{192601} a^{8} - \frac{88795}{192601} a^{7} - \frac{83353}{192601} a^{6} + \frac{75579}{192601} a^{5} - \frac{31302}{192601} a^{4} + \frac{34492}{192601} a^{3} + \frac{95478}{192601} a^{2} + \frac{4397}{192601} a + \frac{1}{192601}$, $\frac{1}{192601} a^{19} + \frac{21969}{192601} a^{17} + \frac{87906}{192601} a^{16} + \frac{76962}{192601} a^{15} + \frac{758}{192601} a^{14} + \frac{25110}{192601} a^{13} + \frac{87244}{192601} a^{12} + \frac{20700}{192601} a^{11} + \frac{44300}{192601} a^{10} - \frac{94901}{192601} a^{9} - \frac{22657}{192601} a^{8} - \frac{53965}{192601} a^{7} + \frac{59017}{192601} a^{6} + \frac{77161}{192601} a^{5} - \frac{40329}{192601} a^{4} + \frac{11141}{192601} a^{3} + \frac{57811}{192601} a^{2} - \frac{73508}{192601} a - \frac{4397}{192601}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{44178756}{192601} a^{19} + \frac{202391951}{192601} a^{18} + \frac{747500583}{192601} a^{17} - \frac{4767816726}{192601} a^{16} - \frac{1114011060}{192601} a^{15} + \frac{36694952193}{192601} a^{14} - \frac{33893590275}{192601} a^{13} - \frac{93106969701}{192601} a^{12} + \frac{132493169760}{192601} a^{11} + \frac{102220489827}{192601} a^{10} - \frac{204889631403}{192601} a^{9} - \frac{39318071136}{192601} a^{8} + \frac{155329081632}{192601} a^{7} - \frac{13773796485}{192601} a^{6} - \frac{55313863005}{192601} a^{5} + \frac{14009629458}{192601} a^{4} + \frac{6850811370}{192601} a^{3} - \frac{2225770689}{192601} a^{2} - \frac{281406231}{192601} a + \frac{104446392}{192601} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1117806.26947 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.4050000.1 x5, 10.2.82012500000000.18 x5

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

Sibling fields

Degree 5 sibling: 5.1.4050000.1
Degree 10 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
3Data not computed
5Data not computed