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Properties

Label 20.12.8938372970...0625.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{10}\cdot 37^{2}\cdot 401^{8}$
Root discriminant $35.28$
Ramified primes $5, 37, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 0, -6237, 0, 18774, 0, -27353, 0, 21832, 0, -10296, 0, 3318, 0, -940, 0, 216, 0, -25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 25*x^18 + 216*x^16 - 940*x^14 + 3318*x^12 - 10296*x^10 + 21832*x^8 - 27353*x^6 + 18774*x^4 - 6237*x^2 + 729)
 
gp: K = bnfinit(x^20 - 25*x^18 + 216*x^16 - 940*x^14 + 3318*x^12 - 10296*x^10 + 21832*x^8 - 27353*x^6 + 18774*x^4 - 6237*x^2 + 729, 1)
 

Normalized defining polynomial

\( x^{20} - 25 x^{18} + 216 x^{16} - 940 x^{14} + 3318 x^{12} - 10296 x^{10} + 21832 x^{8} - 27353 x^{6} + 18774 x^{4} - 6237 x^{2} + 729 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8938372970405474936544619140625=5^{10}\cdot 37^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.28$
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:  $5, 37, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2}$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{12} - \frac{1}{18} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{9} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{13} - \frac{1}{18} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{7}{18} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{16} + \frac{1}{18} a^{12} - \frac{1}{9} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{6} + \frac{2}{9} a^{4} - \frac{1}{2} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{54} a^{17} - \frac{1}{54} a^{15} - \frac{1}{18} a^{13} - \frac{1}{54} a^{11} + \frac{1}{9} a^{9} - \frac{1}{2} a^{8} + \frac{7}{54} a^{5} - \frac{1}{2} a^{4} + \frac{13}{54} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{886074041214} a^{18} + \frac{6386364443}{886074041214} a^{16} - \frac{825991837}{98452671246} a^{14} + \frac{12249636107}{886074041214} a^{12} + \frac{22464249235}{147679006869} a^{10} - \frac{1}{2} a^{9} - \frac{21889034435}{49226335623} a^{8} + \frac{150543678049}{886074041214} a^{6} - \frac{1}{2} a^{5} - \frac{289820820521}{886074041214} a^{4} - \frac{1}{2} a^{3} - \frac{2605626079}{98452671246} a^{2} - \frac{1}{2} a + \frac{1161287009}{5469592847}$, $\frac{1}{2658222123642} a^{19} + \frac{6386364443}{2658222123642} a^{17} - \frac{825991837}{295358013738} a^{15} + \frac{12249636107}{2658222123642} a^{13} + \frac{22464249235}{443037020607} a^{11} - \frac{93004404493}{295358013738} a^{9} + \frac{1036617719263}{2658222123642} a^{7} - \frac{1}{2} a^{6} - \frac{1175894861735}{2658222123642} a^{5} - \frac{1}{2} a^{4} - \frac{25915980851}{147679006869} a^{3} - \frac{1}{2} a^{2} - \frac{3147018829}{32817557082} a - \frac{1}{2}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 331083244.993 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1, 10.6.956707579237.1, 10.6.2989711185115625.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed