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Properties

Label 20.0.95159057403...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 89^{10}$
Root discriminant $31.54$
Ramified primes $5, 89$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![25, -75, 415, -1190, 2746, -5241, 7784, -9101, 9730, -8941, 7077, -5173, 3618, -1983, 1027, -482, 194, -52, 23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 23*x^18 - 52*x^17 + 194*x^16 - 482*x^15 + 1027*x^14 - 1983*x^13 + 3618*x^12 - 5173*x^11 + 7077*x^10 - 8941*x^9 + 9730*x^8 - 9101*x^7 + 7784*x^6 - 5241*x^5 + 2746*x^4 - 1190*x^3 + 415*x^2 - 75*x + 25)
 
gp: K = bnfinit(x^20 - 2*x^19 + 23*x^18 - 52*x^17 + 194*x^16 - 482*x^15 + 1027*x^14 - 1983*x^13 + 3618*x^12 - 5173*x^11 + 7077*x^10 - 8941*x^9 + 9730*x^8 - 9101*x^7 + 7784*x^6 - 5241*x^5 + 2746*x^4 - 1190*x^3 + 415*x^2 - 75*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 23 x^{18} - 52 x^{17} + 194 x^{16} - 482 x^{15} + 1027 x^{14} - 1983 x^{13} + 3618 x^{12} - 5173 x^{11} + 7077 x^{10} - 8941 x^{9} + 9730 x^{8} - 9101 x^{7} + 7784 x^{6} - 5241 x^{5} + 2746 x^{4} - 1190 x^{3} + 415 x^{2} - 75 x + 25 \)

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:  \(951590574034612536651611328125=5^{15}\cdot 89^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.54$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{14} - \frac{3}{20} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{5} a^{10} - \frac{1}{4} a^{9} + \frac{1}{20} a^{8} + \frac{1}{4} a^{7} + \frac{3}{10} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{10} a^{3} - \frac{1}{5} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{20} a^{15} - \frac{1}{5} a^{13} + \frac{1}{20} a^{11} + \frac{3}{20} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{9}{20} a^{7} - \frac{7}{20} a^{6} - \frac{1}{4} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} + \frac{3}{20} a^{2} - \frac{1}{4}$, $\frac{1}{20} a^{16} - \frac{1}{10} a^{13} + \frac{1}{20} a^{12} + \frac{3}{20} a^{11} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} + \frac{3}{20} a^{7} + \frac{9}{20} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{3}{10} a^{2} + \frac{1}{4} a$, $\frac{1}{20} a^{17} - \frac{1}{4} a^{13} + \frac{3}{20} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{9}{20} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{3}{20} a^{2} - \frac{1}{2}$, $\frac{1}{20} a^{18} - \frac{1}{10} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{5} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{7}{20} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{793975221845124403772200} a^{19} - \frac{5488261520355333861201}{793975221845124403772200} a^{18} - \frac{3701286126956376757227}{198493805461281100943050} a^{17} + \frac{98855007722559482159}{15879504436902488075444} a^{16} - \frac{2580827034904530118613}{396987610922562201886100} a^{15} + \frac{93592671136108890593}{198493805461281100943050} a^{14} - \frac{71748592954258519469691}{793975221845124403772200} a^{13} + \frac{11482349509516836359713}{396987610922562201886100} a^{12} - \frac{12746449023241830092399}{198493805461281100943050} a^{11} - \frac{19599654238052601056799}{793975221845124403772200} a^{10} + \frac{46254255230317077808557}{198493805461281100943050} a^{9} - \frac{167528767723479038984293}{793975221845124403772200} a^{8} + \frac{370778043858087956500067}{793975221845124403772200} a^{7} + \frac{42673718139553328821813}{396987610922562201886100} a^{6} - \frac{5032777178332659634043}{19849380546128110094305} a^{5} - \frac{26359726692252350859771}{793975221845124403772200} a^{4} + \frac{66672128390560561126993}{158795044369024880754440} a^{3} + \frac{1031285361217743015479}{158795044369024880754440} a^{2} + \frac{492983175878001294597}{7939752218451244037722} a + \frac{14867433463764785417827}{31759008873804976150888}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( -1 \) (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:  \( 3392623.57291 \) (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

\(\Q(\sqrt{5}) \), 4.0.990125.2, 5.1.990125.1 x5, 10.2.4901737578125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.990125.1
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{20}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$