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Properties

Label 20.0.72246205889...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 109^{10}$
Root discriminant $34.91$
Ramified primes $5, 109$
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, 25, 360, -505, 231, -6763, 9671, 14903, -7755, -2908, 9908, -1396, -3262, 1281, 648, -354, -21, 64, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 8*x^18 + 64*x^17 - 21*x^16 - 354*x^15 + 648*x^14 + 1281*x^13 - 3262*x^12 - 1396*x^11 + 9908*x^10 - 2908*x^9 - 7755*x^8 + 14903*x^7 + 9671*x^6 - 6763*x^5 + 231*x^4 - 505*x^3 + 360*x^2 + 25*x + 25)
 
gp: K = bnfinit(x^20 - 4*x^19 - 8*x^18 + 64*x^17 - 21*x^16 - 354*x^15 + 648*x^14 + 1281*x^13 - 3262*x^12 - 1396*x^11 + 9908*x^10 - 2908*x^9 - 7755*x^8 + 14903*x^7 + 9671*x^6 - 6763*x^5 + 231*x^4 - 505*x^3 + 360*x^2 + 25*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 8 x^{18} + 64 x^{17} - 21 x^{16} - 354 x^{15} + 648 x^{14} + 1281 x^{13} - 3262 x^{12} - 1396 x^{11} + 9908 x^{10} - 2908 x^{9} - 7755 x^{8} + 14903 x^{7} + 9671 x^{6} - 6763 x^{5} + 231 x^{4} - 505 x^{3} + 360 x^{2} + 25 x + 25 \)

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:  \(7224620588965201519805908203125=5^{15}\cdot 109^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.91$
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:  $5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{13} + \frac{1}{10} a^{10} - \frac{1}{10} a^{8} - \frac{2}{5} a^{6} - \frac{1}{10} a^{3} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{15} - \frac{1}{10} a^{13} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{10} a^{4} - \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{16} - \frac{1}{20} a^{15} - \frac{1}{20} a^{14} + \frac{1}{20} a^{13} + \frac{1}{20} a^{12} + \frac{3}{20} a^{10} - \frac{1}{4} a^{9} + \frac{1}{10} a^{8} + \frac{9}{20} a^{6} - \frac{3}{10} a^{5} - \frac{9}{20} a^{4} + \frac{3}{10} a^{3} - \frac{3}{10} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{20} a^{17} + \frac{1}{20} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{9}{20} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{20} a^{2} + \frac{1}{4}$, $\frac{1}{220} a^{18} - \frac{1}{110} a^{17} + \frac{1}{44} a^{16} + \frac{7}{220} a^{15} + \frac{1}{220} a^{14} - \frac{1}{110} a^{13} - \frac{8}{55} a^{12} - \frac{19}{110} a^{11} - \frac{8}{55} a^{10} - \frac{27}{220} a^{9} + \frac{103}{220} a^{8} + \frac{19}{44} a^{7} + \frac{19}{110} a^{6} - \frac{1}{44} a^{5} - \frac{7}{20} a^{4} + \frac{1}{44} a^{3} + \frac{19}{55} a^{2} - \frac{7}{22} a - \frac{21}{44}$, $\frac{1}{86979815590624322200477999211010100} a^{19} - \frac{8577063284084030249192308611791}{43489907795312161100238999605505050} a^{18} - \frac{1620704864452317123198282383129677}{86979815590624322200477999211010100} a^{17} + \frac{205655782482978528006713150701549}{8697981559062432220047799921101010} a^{16} - \frac{2041638151703023002635792954803523}{43489907795312161100238999605505050} a^{15} + \frac{356974729229321016054618625956639}{86979815590624322200477999211010100} a^{14} + \frac{14948706773211654636629821250694931}{86979815590624322200477999211010100} a^{13} + \frac{4259210447267298524249994524013273}{86979815590624322200477999211010100} a^{12} - \frac{5871010670466761535277497227264233}{43489907795312161100238999605505050} a^{11} + \frac{10295469550195834958339400139664151}{43489907795312161100238999605505050} a^{10} - \frac{3914029285857573994021945149369392}{21744953897656080550119499802752525} a^{9} - \frac{17980448847700965912061977893605229}{86979815590624322200477999211010100} a^{8} - \frac{2724846995982767129199232434439122}{21744953897656080550119499802752525} a^{7} + \frac{1544049195639253123585820035705181}{43489907795312161100238999605505050} a^{6} + \frac{3716594541179125648481776378267887}{17395963118124864440095599842202020} a^{5} + \frac{18620998037942993375031695893139931}{43489907795312161100238999605505050} a^{4} + \frac{1057033120424458663326853611297757}{4348990779531216110023899960550505} a^{3} + \frac{1987485011856918714714860584072628}{4348990779531216110023899960550505} a^{2} + \frac{213327921585480016382003593718466}{869798155906243222004779992110101} a - \frac{1652870470762030162160318638820233}{3479192623624972888019119968440404}$

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:  \( 9643655.9171 \) (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

\(\Q(\sqrt{5}) \), 4.0.1485125.1, 5.1.1485125.1 x5, 10.2.11027981328125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.1485125.1
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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$$[\ ]_{4}$
$109$109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$