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Properties

Label 20.0.25553569912...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{23}\cdot 11^{8}$
Root discriminant $16.61$
Ramified primes $5, 11$
Class number $1$
Class group Trivial
Galois group $C_5:F_5$ (as 20T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -50, 250, -825, 2005, -3800, 5800, -7250, 7495, -6490, 4830, -3225, 2010, -1160, 595, -275, 120, -45, 15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 45*x^17 + 120*x^16 - 275*x^15 + 595*x^14 - 1160*x^13 + 2010*x^12 - 3225*x^11 + 4830*x^10 - 6490*x^9 + 7495*x^8 - 7250*x^7 + 5800*x^6 - 3800*x^5 + 2005*x^4 - 825*x^3 + 250*x^2 - 50*x + 5)
 
gp: K = bnfinit(x^20 - 5*x^19 + 15*x^18 - 45*x^17 + 120*x^16 - 275*x^15 + 595*x^14 - 1160*x^13 + 2010*x^12 - 3225*x^11 + 4830*x^10 - 6490*x^9 + 7495*x^8 - 7250*x^7 + 5800*x^6 - 3800*x^5 + 2005*x^4 - 825*x^3 + 250*x^2 - 50*x + 5, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 15 x^{18} - 45 x^{17} + 120 x^{16} - 275 x^{15} + 595 x^{14} - 1160 x^{13} + 2010 x^{12} - 3225 x^{11} + 4830 x^{10} - 6490 x^{9} + 7495 x^{8} - 7250 x^{7} + 5800 x^{6} - 3800 x^{5} + 2005 x^{4} - 825 x^{3} + 250 x^{2} - 50 x + 5 \)

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:  \(2555356991291046142578125=5^{23}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.61$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{16} - \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{1062} a^{18} - \frac{35}{531} a^{17} + \frac{14}{531} a^{16} - \frac{110}{531} a^{15} - \frac{22}{531} a^{14} - \frac{35}{177} a^{13} - \frac{20}{177} a^{12} + \frac{212}{531} a^{11} - \frac{214}{531} a^{10} + \frac{49}{177} a^{9} + \frac{13}{531} a^{8} + \frac{155}{531} a^{7} - \frac{29}{59} a^{6} - \frac{14}{59} a^{5} - \frac{34}{531} a^{4} - \frac{361}{1062} a^{3} + \frac{259}{531} a^{2} - \frac{43}{177} a + \frac{34}{531}$, $\frac{1}{98766} a^{19} - \frac{1}{32922} a^{18} + \frac{83}{3658} a^{17} - \frac{203}{5487} a^{16} + \frac{1975}{32922} a^{15} + \frac{14365}{98766} a^{14} - \frac{3250}{16461} a^{13} - \frac{7525}{49383} a^{12} - \frac{49015}{98766} a^{11} + \frac{2801}{49383} a^{10} + \frac{24199}{49383} a^{9} + \frac{641}{10974} a^{8} + \frac{13841}{49383} a^{7} + \frac{2704}{5487} a^{6} - \frac{24917}{98766} a^{5} + \frac{8627}{32922} a^{4} + \frac{757}{98766} a^{3} + \frac{17224}{49383} a^{2} + \frac{20065}{49383} a + \frac{47567}{98766}$

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:  \( -\frac{32970}{1829} a^{19} + \frac{1721021}{16461} a^{18} - \frac{5150663}{16461} a^{17} + \frac{15031040}{16461} a^{16} - \frac{82469191}{32922} a^{15} + \frac{188633467}{32922} a^{14} - \frac{67454492}{5487} a^{13} + \frac{133242175}{5487} a^{12} - \frac{1375307855}{32922} a^{11} + \frac{1096050242}{16461} a^{10} - \frac{547345688}{5487} a^{9} + \frac{4405634855}{32922} a^{8} - \frac{2511799585}{16461} a^{7} + \frac{262940540}{1829} a^{6} - \frac{403490937}{3658} a^{5} + \frac{1122123980}{16461} a^{4} - \frac{544702865}{16461} a^{3} + \frac{394283975}{32922} a^{2} - \frac{16006240}{5487} a + \frac{12224003}{32922} \) (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:  \( 56436.2432248 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:F_5$ (as 20T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 10 conjugacy class representatives for $C_5:F_5$
Character table for $C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.714892578125.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 sibling: data not computed
Degree 25 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$