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Properties

Label 21.21.2895190882...6944.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{32}$
Root discriminant $35.14$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:C_3$ (as 21T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -14, 49, 161, -1547, 3514, 392, -13533, 16660, 8897, -32431, 14819, 16856, -18536, 1861, 4732, -1848, -245, 238, -21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 21*x^19 + 238*x^18 - 245*x^17 - 1848*x^16 + 4732*x^15 + 1861*x^14 - 18536*x^13 + 16856*x^12 + 14819*x^11 - 32431*x^10 + 8897*x^9 + 16660*x^8 - 13533*x^7 + 392*x^6 + 3514*x^5 - 1547*x^4 + 161*x^3 + 49*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^21 - 7*x^20 - 21*x^19 + 238*x^18 - 245*x^17 - 1848*x^16 + 4732*x^15 + 1861*x^14 - 18536*x^13 + 16856*x^12 + 14819*x^11 - 32431*x^10 + 8897*x^9 + 16660*x^8 - 13533*x^7 + 392*x^6 + 3514*x^5 - 1547*x^4 + 161*x^3 + 49*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 21 x^{19} + 238 x^{18} - 245 x^{17} - 1848 x^{16} + 4732 x^{15} + 1861 x^{14} - 18536 x^{13} + 16856 x^{12} + 14819 x^{11} - 32431 x^{10} + 8897 x^{9} + 16660 x^{8} - 13533 x^{7} + 392 x^{6} + 3514 x^{5} - 1547 x^{4} + 161 x^{3} + 49 x^{2} - 14 x + 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(289519088236998333905056353746944=2^{18}\cdot 7^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.14$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2339} a^{18} + \frac{637}{2339} a^{17} + \frac{241}{2339} a^{16} - \frac{264}{2339} a^{15} - \frac{258}{2339} a^{14} + \frac{983}{2339} a^{13} + \frac{1149}{2339} a^{12} - \frac{300}{2339} a^{11} + \frac{406}{2339} a^{10} - \frac{668}{2339} a^{9} + \frac{652}{2339} a^{8} - \frac{1072}{2339} a^{7} + \frac{595}{2339} a^{6} + \frac{1044}{2339} a^{5} + \frac{139}{2339} a^{4} + \frac{181}{2339} a^{3} - \frac{472}{2339} a^{2} - \frac{655}{2339} a + \frac{1}{2339}$, $\frac{1}{423359} a^{19} - \frac{90}{423359} a^{18} + \frac{168672}{423359} a^{17} + \frac{42056}{423359} a^{16} - \frac{9484}{423359} a^{15} - \frac{87453}{423359} a^{14} + \frac{44344}{423359} a^{13} - \frac{61414}{423359} a^{12} + \frac{131963}{423359} a^{11} + \frac{15257}{423359} a^{10} - \frac{93784}{423359} a^{9} + \frac{177505}{423359} a^{8} + \frac{33798}{423359} a^{7} + \frac{8211}{423359} a^{6} - \frac{108607}{423359} a^{5} + \frac{196181}{423359} a^{4} + \frac{66756}{423359} a^{3} - \frac{186125}{423359} a^{2} + \frac{85573}{423359} a + \frac{127918}{423359}$, $\frac{1}{76627979} a^{20} + \frac{42}{76627979} a^{19} + \frac{2037}{76627979} a^{18} + \frac{37611396}{76627979} a^{17} - \frac{9316020}{76627979} a^{16} + \frac{1413669}{76627979} a^{15} - \frac{25339255}{76627979} a^{14} + \frac{38252353}{76627979} a^{13} - \frac{25758478}{76627979} a^{12} - \frac{489695}{76627979} a^{11} - \frac{13070823}{76627979} a^{10} + \frac{29787302}{76627979} a^{9} + \frac{3425767}{76627979} a^{8} + \frac{27694924}{76627979} a^{7} - \frac{4738744}{76627979} a^{6} + \frac{18194259}{76627979} a^{5} - \frac{23066632}{76627979} a^{4} + \frac{22527553}{76627979} a^{3} + \frac{721866}{76627979} a^{2} + \frac{31079792}{76627979} a + \frac{18000498}{76627979}$

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:  $20$
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:  \( 3144548714.36 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:C_3$ (as 21T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.18078415936.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.18078415936.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
7Data not computed