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Properties

Label 20.0.53371420399...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 53^{10}$
Root discriminant $24.34$
Ramified primes $5, 53$
Class number $2$
Class group $[2]$
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, 1225, 2325, 2240, -1385, 3925, 5430, 1556, -1279, 620, 840, -199, -40, 54, 50, -19, -5, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 5*x^17 - 19*x^16 + 50*x^15 + 54*x^14 - 40*x^13 - 199*x^12 + 840*x^11 + 620*x^10 - 1279*x^9 + 1556*x^8 + 5430*x^7 + 3925*x^6 - 1385*x^5 + 2240*x^4 + 2325*x^3 + 1225*x^2 + 25*x + 25)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 5*x^17 - 19*x^16 + 50*x^15 + 54*x^14 - 40*x^13 - 199*x^12 + 840*x^11 + 620*x^10 - 1279*x^9 + 1556*x^8 + 5430*x^7 + 3925*x^6 - 1385*x^5 + 2240*x^4 + 2325*x^3 + 1225*x^2 + 25*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 5 x^{17} - 19 x^{16} + 50 x^{15} + 54 x^{14} - 40 x^{13} - 199 x^{12} + 840 x^{11} + 620 x^{10} - 1279 x^{9} + 1556 x^{8} + 5430 x^{7} + 3925 x^{6} - 1385 x^{5} + 2240 x^{4} + 2325 x^{3} + 1225 x^{2} + 25 x + 25 \)

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:  \(5337142039963166778564453125=5^{15}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.34$
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:  $5, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{10} a^{16} + \frac{1}{10} a^{15} + \frac{1}{10} a^{12} + \frac{2}{5} a^{10} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{15} + \frac{1}{10} a^{13} - \frac{1}{10} a^{12} + \frac{2}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} + \frac{2}{5} a^{8} - \frac{1}{2} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{18} + \frac{1}{10} a^{15} + \frac{1}{10} a^{14} - \frac{1}{10} a^{13} + \frac{1}{10} a^{11} - \frac{1}{2} a^{10} + \frac{2}{5} a^{9} + \frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1109624125897833818503322272586484550} a^{19} + \frac{25277064716404198236814452398582663}{554812062948916909251661136293242275} a^{18} + \frac{2211302209396121618505522296969021}{221924825179566763700664454517296910} a^{17} - \frac{2985437273225229592472834033646329}{221924825179566763700664454517296910} a^{16} - \frac{77617666923965521013956108174978072}{554812062948916909251661136293242275} a^{15} - \frac{15621440729532865925593296106844169}{110962412589783381850332227258648455} a^{14} - \frac{98269321072279400910902475417347178}{554812062948916909251661136293242275} a^{13} - \frac{9469023098796717679699255248380809}{44384965035913352740132890903459382} a^{12} + \frac{57370654206334931646667951129943133}{554812062948916909251661136293242275} a^{11} - \frac{9198580663172813067695369802715176}{22192482517956676370066445451729691} a^{10} + \frac{6288749752104712579107128146852461}{221924825179566763700664454517296910} a^{9} - \frac{107529354471926795127737353337762742}{554812062948916909251661136293242275} a^{8} - \frac{104814870533440827125661252101813192}{554812062948916909251661136293242275} a^{7} - \frac{7256323117176685618419655515913859}{44384965035913352740132890903459382} a^{6} + \frac{24883537927648188413878742163740221}{110962412589783381850332227258648455} a^{5} - \frac{57333215948427563518844250510262897}{221924825179566763700664454517296910} a^{4} - \frac{83237293360162516435646188204376111}{221924825179566763700664454517296910} a^{3} - \frac{8949325509928637214223884360491575}{44384965035913352740132890903459382} a^{2} + \frac{162623925591655705229516728683691}{752287542981582249832760862770498} a + \frac{11938344268209793265074191380940755}{44384965035913352740132890903459382}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 367558.874812 \)
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

\(\Q(\sqrt{5}) \), 4.0.351125.1, 5.1.351125.1 x5, 10.2.616443828125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.351125.1
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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$