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Properties

Label 20.0.13792736767...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{31}\cdot 7^{10}$
Root discriminant $64.12$
Ramified primes $2, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22598019920, 0, 4035360700, 0, -1008840175, 0, -247062900, 0, 5882450, 0, 4537890, 0, 324135, 0, 10290, 0, 735, 0, 35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 35*x^18 + 735*x^16 + 10290*x^14 + 324135*x^12 + 4537890*x^10 + 5882450*x^8 - 247062900*x^6 - 1008840175*x^4 + 4035360700*x^2 + 22598019920)
 
gp: K = bnfinit(x^20 + 35*x^18 + 735*x^16 + 10290*x^14 + 324135*x^12 + 4537890*x^10 + 5882450*x^8 - 247062900*x^6 - 1008840175*x^4 + 4035360700*x^2 + 22598019920, 1)
 

Normalized defining polynomial

\( x^{20} + 35 x^{18} + 735 x^{16} + 10290 x^{14} + 324135 x^{12} + 4537890 x^{10} + 5882450 x^{8} - 247062900 x^{6} - 1008840175 x^{4} + 4035360700 x^{2} + 22598019920 \)

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:  \(1379273676757812500000000000000000000=2^{20}\cdot 5^{31}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.12$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{16807} a^{10}$, $\frac{1}{16807} a^{11}$, $\frac{1}{117649} a^{12}$, $\frac{1}{117649} a^{13}$, $\frac{1}{823543} a^{14}$, $\frac{1}{823543} a^{15}$, $\frac{1}{17294403} a^{16} + \frac{1}{352947} a^{12} + \frac{1}{50421} a^{10} - \frac{1}{7203} a^{8} + \frac{1}{147} a^{4} + \frac{1}{21} a^{2} + \frac{1}{3}$, $\frac{1}{34588806} a^{17} - \frac{1}{1647086} a^{15} + \frac{1}{705894} a^{13} - \frac{1}{50421} a^{11} - \frac{1}{14406} a^{9} - \frac{1}{147} a^{5} - \frac{1}{21} a^{3} + \frac{1}{6} a$, $\frac{1}{864323285165250036} a^{18} + \frac{2036262541}{123474755023607148} a^{16} + \frac{2475704179}{17639250717658164} a^{14} + \frac{3331366495}{1259946479832726} a^{12} + \frac{92995991}{17142128977316} a^{10} - \frac{2800690631}{25713193465974} a^{8} - \frac{1484233081}{3673313352282} a^{6} - \frac{1605116785}{262379525163} a^{4} + \frac{6185082881}{149931157236} a^{2} + \frac{717859552}{5354684187}$, $\frac{1}{1728646570330500072} a^{19} + \frac{2036262541}{246949510047214296} a^{17} - \frac{18943032569}{35278501435316328} a^{15} + \frac{3331366495}{2519892959665452} a^{13} - \frac{6488606979}{239989805682424} a^{11} + \frac{7908677743}{51426386931948} a^{9} + \frac{9225135293}{7346626704564} a^{7} - \frac{1605116785}{524759050326} a^{5} + \frac{6185082881}{299862314472} a^{3} - \frac{4636824635}{10709368374} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.98000.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$