Properties

Label 21.3.490...767.1
Degree $21$
Signature $[3, 9]$
Discriminant $-4.901\times 10^{27}$
Root discriminant $20.82$
Ramified primes $7, 193327$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 21T56

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 2*x^19 + 9*x^18 - 8*x^17 - 16*x^16 + 40*x^15 + 25*x^14 - 4*x^13 - 178*x^12 - 112*x^11 + 286*x^10 + 394*x^9 - 392*x^8 - 175*x^7 + 201*x^6 - 68*x^5 - 36*x^4 + 30*x^3 - 4*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^21 - x^20 - 2*x^19 + 9*x^18 - 8*x^17 - 16*x^16 + 40*x^15 + 25*x^14 - 4*x^13 - 178*x^12 - 112*x^11 + 286*x^10 + 394*x^9 - 392*x^8 - 175*x^7 + 201*x^6 - 68*x^5 - 36*x^4 + 30*x^3 - 4*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -4, 30, -36, -68, 201, -175, -392, 394, 286, -112, -178, -4, 25, 40, -16, -8, 9, -2, -1, 1]);
 

\( x^{21} - x^{20} - 2 x^{19} + 9 x^{18} - 8 x^{17} - 16 x^{16} + 40 x^{15} + 25 x^{14} - 4 x^{13} - 178 x^{12} - 112 x^{11} + 286 x^{10} + 394 x^{9} - 392 x^{8} - 175 x^{7} + 201 x^{6} - 68 x^{5} - 36 x^{4} + 30 x^{3} - 4 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-4900609474961156001559784767\)\(\medspace = -\,7^{14}\cdot 193327^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.82$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 193327$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $3$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{17} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{852374661690369705667161} a^{20} - \frac{37206227685349432218735}{284124887230123235222387} a^{19} + \frac{28400824817952502765440}{284124887230123235222387} a^{18} - \frac{13780621895031325588738}{284124887230123235222387} a^{17} + \frac{61682550390744633237293}{852374661690369705667161} a^{16} - \frac{353934655569229934974865}{852374661690369705667161} a^{15} + \frac{211891158807041184903164}{852374661690369705667161} a^{14} - \frac{409455791776301474388844}{852374661690369705667161} a^{13} - \frac{75605433961189884678952}{852374661690369705667161} a^{12} + \frac{35684403822527476829982}{284124887230123235222387} a^{11} - \frac{207638439644003725936438}{852374661690369705667161} a^{10} - \frac{273603453027009074476673}{852374661690369705667161} a^{9} + \frac{73809348701586405728637}{284124887230123235222387} a^{8} + \frac{156048305230993844754983}{852374661690369705667161} a^{7} + \frac{35451996633254066320589}{284124887230123235222387} a^{6} - \frac{9809087718141362950026}{284124887230123235222387} a^{5} - \frac{115231107106993609580344}{852374661690369705667161} a^{4} - \frac{159253211615910080434012}{852374661690369705667161} a^{3} + \frac{374694091070485556627843}{852374661690369705667161} a^{2} - \frac{247680541936135807642024}{852374661690369705667161} a + \frac{221238723822791693315399}{852374661690369705667161}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 299129.54193 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 299129.54193 \cdot 1}{2\sqrt{4900609474961156001559784767}}\approx 0.26086334857$ (assuming GRH)

Galois group

21T56:

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

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.193327.1

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 $21$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ $15{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R $21$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
193327Data not computed