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Properties

Label 21.21.4807893938...4513.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{14}\cdot 577^{9}$
Root discriminant $55.82$
Ramified primes $7, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times D_7$ (as 21T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 7, 599, -214, -15878, 19398, 54968, -67381, -80247, 91976, 60287, -63963, -25116, 24849, 5869, -5517, -739, 679, 45, -42, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 42*x^19 + 45*x^18 + 679*x^17 - 739*x^16 - 5517*x^15 + 5869*x^14 + 24849*x^13 - 25116*x^12 - 63963*x^11 + 60287*x^10 + 91976*x^9 - 80247*x^8 - 67381*x^7 + 54968*x^6 + 19398*x^5 - 15878*x^4 - 214*x^3 + 599*x^2 + 7*x - 1)
 
gp: K = bnfinit(x^21 - x^20 - 42*x^19 + 45*x^18 + 679*x^17 - 739*x^16 - 5517*x^15 + 5869*x^14 + 24849*x^13 - 25116*x^12 - 63963*x^11 + 60287*x^10 + 91976*x^9 - 80247*x^8 - 67381*x^7 + 54968*x^6 + 19398*x^5 - 15878*x^4 - 214*x^3 + 599*x^2 + 7*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 42 x^{19} + 45 x^{18} + 679 x^{17} - 739 x^{16} - 5517 x^{15} + 5869 x^{14} + 24849 x^{13} - 25116 x^{12} - 63963 x^{11} + 60287 x^{10} + 91976 x^{9} - 80247 x^{8} - 67381 x^{7} + 54968 x^{6} + 19398 x^{5} - 15878 x^{4} - 214 x^{3} + 599 x^{2} + 7 x - 1 \)

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:  \(4807893938270387570039361667473274513=7^{14}\cdot 577^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.82$
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:  $7, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{41} a^{19} + \frac{17}{41} a^{18} + \frac{7}{41} a^{17} - \frac{16}{41} a^{16} - \frac{14}{41} a^{15} + \frac{5}{41} a^{14} + \frac{16}{41} a^{13} - \frac{7}{41} a^{12} - \frac{12}{41} a^{11} + \frac{1}{41} a^{10} - \frac{17}{41} a^{9} - \frac{13}{41} a^{8} + \frac{7}{41} a^{7} + \frac{13}{41} a^{6} + \frac{16}{41} a^{5} + \frac{9}{41} a^{4} - \frac{9}{41} a^{3} + \frac{15}{41} a^{2} - \frac{9}{41} a - \frac{15}{41}$, $\frac{1}{5679038659914732157253659708613} a^{20} - \frac{10840763106943312344809133017}{5679038659914732157253659708613} a^{19} - \frac{1959920471601085449660496842573}{5679038659914732157253659708613} a^{18} - \frac{2394951428210498514742063015071}{5679038659914732157253659708613} a^{17} - \frac{1954296273972732584098892386399}{5679038659914732157253659708613} a^{16} + \frac{1522416888458515522129135518332}{5679038659914732157253659708613} a^{15} + \frac{2445765032610662392140456006811}{5679038659914732157253659708613} a^{14} + \frac{109857576034746689221797400753}{5679038659914732157253659708613} a^{13} + \frac{2348944973934994493599677262292}{5679038659914732157253659708613} a^{12} + \frac{2603636397793355162389263008258}{5679038659914732157253659708613} a^{11} - \frac{1918578740173746912650229208817}{5679038659914732157253659708613} a^{10} - \frac{138626549351956045516655495453}{5679038659914732157253659708613} a^{9} + \frac{2364292050002653220690794657014}{5679038659914732157253659708613} a^{8} - \frac{1063483783022862205232616156476}{5679038659914732157253659708613} a^{7} - \frac{1611582094100525510528171826971}{5679038659914732157253659708613} a^{6} + \frac{731117543324181744430839825463}{5679038659914732157253659708613} a^{5} + \frac{1823955059539027518963383064208}{5679038659914732157253659708613} a^{4} - \frac{2045500731215844883069247129739}{5679038659914732157253659708613} a^{3} - \frac{1821610339989678334526180035767}{5679038659914732157253659708613} a^{2} - \frac{442291460669373706454021659619}{5679038659914732157253659708613} a - \frac{533400164411091294440736250193}{5679038659914732157253659708613}$

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:  \( 524399167092 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_7$ (as 21T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 15 conjugacy class representatives for $C_3\times D_7$
Character table for $C_3\times D_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.192100033.1

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

Sibling fields

Galois closure: 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 $21$ $21$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $21$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ $21$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
577Data not computed