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Properties

Label 20.4.11043582468...7856.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{52}\cdot 31^{4}\cdot 227^{4}$
Root discriminant $35.66$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 330, 0, 1589, 0, -140, 0, -5426, 0, -6312, 0, -2814, 0, -452, 0, 25, 0, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 14*x^18 + 25*x^16 - 452*x^14 - 2814*x^12 - 6312*x^10 - 5426*x^8 - 140*x^6 + 1589*x^4 + 330*x^2 + 1)
 
gp: K = bnfinit(x^20 + 14*x^18 + 25*x^16 - 452*x^14 - 2814*x^12 - 6312*x^10 - 5426*x^8 - 140*x^6 + 1589*x^4 + 330*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 14 x^{18} + 25 x^{16} - 452 x^{14} - 2814 x^{12} - 6312 x^{10} - 5426 x^{8} - 140 x^{6} + 1589 x^{4} + 330 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11043582468423783283165921017856=2^{52}\cdot 31^{4}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.66$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{3}{16}$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{13}{32} a + \frac{3}{32}$, $\frac{1}{924168032} a^{18} - \frac{14929571}{924168032} a^{16} - \frac{3029547}{57760502} a^{14} - \frac{191955}{7452968} a^{12} - \frac{15212363}{462084016} a^{10} - \frac{27407801}{462084016} a^{8} + \frac{44330485}{231042008} a^{6} - \frac{13922355}{115521004} a^{4} - \frac{1}{2} a^{3} + \frac{376424945}{924168032} a^{2} - \frac{1}{2} a - \frac{330527423}{924168032}$, $\frac{1}{1848336064} a^{19} - \frac{1}{1848336064} a^{18} - \frac{14929571}{1848336064} a^{17} + \frac{14929571}{1848336064} a^{16} - \frac{3029547}{115521004} a^{15} + \frac{3029547}{115521004} a^{14} - \frac{191955}{14905936} a^{13} + \frac{191955}{14905936} a^{12} + \frac{42548139}{924168032} a^{11} - \frac{42548139}{924168032} a^{10} - \frac{85168303}{924168032} a^{9} + \frac{85168303}{924168032} a^{8} - \frac{71190519}{462084016} a^{7} + \frac{71190519}{462084016} a^{6} + \frac{36359199}{115521004} a^{5} - \frac{36359199}{115521004} a^{4} - \frac{201180075}{1848336064} a^{3} + \frac{201180075}{1848336064} a^{2} - \frac{908132443}{1848336064} a + \frac{908132443}{1848336064}$

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:  $11$
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:  \( 144631047.555 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Sibling fields

Degree 20 sibling: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.8.26.3$x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
2.12.26.27$x^{12} - 2 x^{10} + 4 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.10.0.1$x^{10} - x + 11$$1$$10$$0$$C_{10}$$[\ ]^{10}$
227Data not computed