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Properties

Label 21.15.9019481696...4807.1
Degree $21$
Signature $[15, 3]$
Discriminant $-\,7^{14}\cdot 23^{3}\cdot 239^{3}\cdot 431^{3}$
Root discriminant $29.79$
Ramified primes $7, 23, 239, 431$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T56

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, -31, 5, 272, -26, -566, -233, 150, 641, 307, -477, 182, 149, -350, -16, 135, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 20*x^19 + 135*x^17 - 16*x^16 - 350*x^15 + 149*x^14 + 182*x^13 - 477*x^12 + 307*x^11 + 641*x^10 + 150*x^9 - 233*x^8 - 566*x^7 - 26*x^6 + 272*x^5 + 5*x^4 - 31*x^3 + 7*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^21 - 20*x^19 + 135*x^17 - 16*x^16 - 350*x^15 + 149*x^14 + 182*x^13 - 477*x^12 + 307*x^11 + 641*x^10 + 150*x^9 - 233*x^8 - 566*x^7 - 26*x^6 + 272*x^5 + 5*x^4 - 31*x^3 + 7*x^2 - 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 20 x^{19} + 135 x^{17} - 16 x^{16} - 350 x^{15} + 149 x^{14} + 182 x^{13} - 477 x^{12} + 307 x^{11} + 641 x^{10} + 150 x^{9} - 233 x^{8} - 566 x^{7} - 26 x^{6} + 272 x^{5} + 5 x^{4} - 31 x^{3} + 7 x^{2} - 2 x - 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-9019481696572641947795544614807=-\,7^{14}\cdot 23^{3}\cdot 239^{3}\cdot 431^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.79$
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:  $7, 23, 239, 431$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{43480328727566562334733} a^{20} - \frac{19511862173060278758012}{43480328727566562334733} a^{19} + \frac{10324280504272507671164}{43480328727566562334733} a^{18} + \frac{2917404655226904672091}{43480328727566562334733} a^{17} + \frac{1660943326837312375553}{43480328727566562334733} a^{16} - \frac{3577486643971422658727}{43480328727566562334733} a^{15} - \frac{452696080077384189359}{1499321680260915942577} a^{14} + \frac{17153848660850035375487}{43480328727566562334733} a^{13} + \frac{8089947734116738266801}{43480328727566562334733} a^{12} - \frac{17815250120820090514495}{43480328727566562334733} a^{11} - \frac{13022509446934936817716}{43480328727566562334733} a^{10} + \frac{1667279086503356371153}{43480328727566562334733} a^{9} + \frac{14572306695118037442194}{43480328727566562334733} a^{8} - \frac{16611454494888335026170}{43480328727566562334733} a^{7} - \frac{15634700091503183100171}{43480328727566562334733} a^{6} - \frac{17346938062931767699817}{43480328727566562334733} a^{5} - \frac{9943591369520444950239}{43480328727566562334733} a^{4} - \frac{9078876537010071507633}{43480328727566562334733} a^{3} - \frac{732082737703405994837}{43480328727566562334733} a^{2} + \frac{128084911589816894444}{1499321680260915942577} a + \frac{527007668253826887143}{43480328727566562334733}$

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:  $17$
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:  \( 126151281.589 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T56:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15120
The 45 conjugacy class representatives for t21n56
Character table for t21n56 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.5.2369207.1

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

Sibling fields

Degree 42 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 $21$ $21$ $15{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R $15{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ $15{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$23$23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
239Data not computed
431Data not computed