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Properties

Label 21.21.3199336816...6896.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 73^{14}$
Root discriminant $31.64$
Ramified primes $2, 73$
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, -10, 399, -1399, -9947, 58226, -74030, -52267, 164304, -52599, -103109, 77939, 16294, -31996, 4781, 5310, -1890, -267, 208, -13, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1)
 
gp: K = bnfinit(x^21 - 7*x^20 - 13*x^19 + 208*x^18 - 267*x^17 - 1890*x^16 + 5310*x^15 + 4781*x^14 - 31996*x^13 + 16294*x^12 + 77939*x^11 - 103109*x^10 - 52599*x^9 + 164304*x^8 - 52267*x^7 - 74030*x^6 + 58226*x^5 - 9947*x^4 - 1399*x^3 + 399*x^2 - 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 13 x^{19} + 208 x^{18} - 267 x^{17} - 1890 x^{16} + 5310 x^{15} + 4781 x^{14} - 31996 x^{13} + 16294 x^{12} + 77939 x^{11} - 103109 x^{10} - 52599 x^{9} + 164304 x^{8} - 52267 x^{7} - 74030 x^{6} + 58226 x^{5} - 9947 x^{4} - 1399 x^{3} + 399 x^{2} - 10 x - 1 \)

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:  \(31993368160435329741277209296896=2^{18}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.64$
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:  $2, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{17} - \frac{1}{5} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{19} - \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{1618089476600365} a^{20} + \frac{58123111048383}{1618089476600365} a^{19} + \frac{3734999696176}{1618089476600365} a^{18} - \frac{515714038752107}{1618089476600365} a^{17} + \frac{375135922018768}{1618089476600365} a^{16} + \frac{116618451789255}{323617895320073} a^{15} + \frac{376267587928901}{1618089476600365} a^{14} - \frac{140697310776285}{323617895320073} a^{13} - \frac{766626976463449}{1618089476600365} a^{12} - \frac{753501798021011}{1618089476600365} a^{11} - \frac{286446107718676}{1618089476600365} a^{10} + \frac{130652890560010}{323617895320073} a^{9} - \frac{224750701894986}{1618089476600365} a^{8} - \frac{602348861250697}{1618089476600365} a^{7} + \frac{226197888251838}{1618089476600365} a^{6} - \frac{202470394756069}{1618089476600365} a^{5} + \frac{208504001442222}{1618089476600365} a^{4} - \frac{122526706673186}{1618089476600365} a^{3} - \frac{110091881694321}{323617895320073} a^{2} - \frac{514522742215396}{1618089476600365} a - \frac{485200243909626}{1618089476600365}$

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:  \( 802783691.084 \) (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

3.3.5329.1, 7.7.1817487424.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.1817487424.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$73$73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$