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Properties

Label 21.1.25039658650...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{38}\cdot 3^{22}\cdot 5^{38}\cdot 7^{20}$
Root discriminant $1300.83$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T163

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-400, 420, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 420*x - 400)
 
gp: K = bnfinit(x^21 + 420*x - 400, 1)
 

Normalized defining polynomial

\( x^{21} + 420 x - 400 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(250396586502256393773476760900000000000000000000000000000000000000=2^{38}\cdot 3^{22}\cdot 5^{38}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1300.83$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{18} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{19} - \frac{1}{2} a^{9}$, $\frac{1}{140} a^{20} - \frac{1}{28} a^{19} - \frac{1}{14} a^{18} + \frac{3}{28} a^{17} - \frac{1}{28} a^{16} - \frac{1}{14} a^{15} + \frac{3}{28} a^{14} - \frac{1}{28} a^{13} - \frac{1}{14} a^{12} + \frac{3}{28} a^{11} + \frac{3}{14} a^{10} + \frac{3}{7} a^{9} + \frac{5}{14} a^{8} + \frac{3}{14} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{14} a^{4} - \frac{1}{14} a^{3} - \frac{1}{7} a^{2} + \frac{3}{14} a + \frac{3}{7}$

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:  $10$
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:  \( 19330421884200000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T163:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 25545471085854720000
The 408 conjugacy class representatives for A21 are not computed
Character table for A21 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R R $18{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $19{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $17{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
3Data not computed
5Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.7.8.4$x^{7} + 21 x^{2} + 7$$7$$1$$8$$F_7$$[4/3]_{3}^{2}$
7.7.7.6$x^{7} + 28 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$