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Properties

Label 21.3.48958786234...5327.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 29^{18}$
Root discriminant $86.62$
Ramified primes $7, 29$
Class number $49$ (GRH)
Class group $[7, 7]$ (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![15197, -225533, 614026, -486339, -1534666, 764470, 426315, -1168123, -139668, 228083, -158805, -65282, -26711, -14133, 3148, -2200, 545, 116, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 116*x^17 + 545*x^16 - 2200*x^15 + 3148*x^14 - 14133*x^13 - 26711*x^12 - 65282*x^11 - 158805*x^10 + 228083*x^9 - 139668*x^8 - 1168123*x^7 + 426315*x^6 + 764470*x^5 - 1534666*x^4 - 486339*x^3 + 614026*x^2 - 225533*x + 15197)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 116*x^17 + 545*x^16 - 2200*x^15 + 3148*x^14 - 14133*x^13 - 26711*x^12 - 65282*x^11 - 158805*x^10 + 228083*x^9 - 139668*x^8 - 1168123*x^7 + 426315*x^6 + 764470*x^5 - 1534666*x^4 - 486339*x^3 + 614026*x^2 - 225533*x + 15197, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 116 x^{17} + 545 x^{16} - 2200 x^{15} + 3148 x^{14} - 14133 x^{13} - 26711 x^{12} - 65282 x^{11} - 158805 x^{10} + 228083 x^{9} - 139668 x^{8} - 1168123 x^{7} + 426315 x^{6} + 764470 x^{5} - 1534666 x^{4} - 486339 x^{3} + 614026 x^{2} - 225533 x + 15197 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-48958786234221774012025092444897160075327=-\,7^{17}\cdot 29^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.62$
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:  $7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{43} a^{16} + \frac{8}{43} a^{15} + \frac{6}{43} a^{13} + \frac{15}{43} a^{12} - \frac{12}{43} a^{11} - \frac{1}{43} a^{10} + \frac{8}{43} a^{9} - \frac{3}{43} a^{8} - \frac{3}{43} a^{7} + \frac{15}{43} a^{6} + \frac{2}{43} a^{5} + \frac{2}{43} a^{4} + \frac{17}{43} a^{3} - \frac{19}{43} a^{2} + \frac{17}{43} a - \frac{10}{43}$, $\frac{1}{43} a^{17} - \frac{21}{43} a^{15} + \frac{6}{43} a^{14} + \frac{10}{43} a^{13} - \frac{3}{43} a^{12} + \frac{9}{43} a^{11} + \frac{16}{43} a^{10} + \frac{19}{43} a^{9} + \frac{21}{43} a^{8} - \frac{4}{43} a^{7} + \frac{11}{43} a^{6} - \frac{14}{43} a^{5} + \frac{1}{43} a^{4} + \frac{17}{43} a^{3} - \frac{3}{43} a^{2} - \frac{17}{43} a - \frac{6}{43}$, $\frac{1}{16211} a^{18} - \frac{147}{16211} a^{17} + \frac{109}{16211} a^{16} - \frac{294}{1247} a^{15} - \frac{4011}{16211} a^{14} + \frac{3048}{16211} a^{13} - \frac{7705}{16211} a^{12} - \frac{5361}{16211} a^{11} - \frac{8010}{16211} a^{10} + \frac{160}{16211} a^{9} + \frac{2453}{16211} a^{8} - \frac{1812}{16211} a^{7} + \frac{3716}{16211} a^{6} - \frac{6754}{16211} a^{5} - \frac{6406}{16211} a^{4} - \frac{8075}{16211} a^{3} - \frac{3336}{16211} a^{2} + \frac{7842}{16211} a + \frac{292}{1247}$, $\frac{1}{16211} a^{19} - \frac{11}{16211} a^{17} + \frac{137}{16211} a^{16} + \frac{4933}{16211} a^{15} - \frac{3727}{16211} a^{14} - \frac{739}{16211} a^{13} - \frac{5488}{16211} a^{12} - \frac{4000}{16211} a^{11} + \frac{5339}{16211} a^{10} - \frac{2679}{16211} a^{9} + \frac{76}{377} a^{8} - \frac{4403}{16211} a^{7} - \frac{4890}{16211} a^{6} + \frac{5084}{16211} a^{5} + \frac{4053}{16211} a^{4} + \frac{7368}{16211} a^{3} + \frac{6419}{16211} a^{2} + \frac{2573}{16211} a - \frac{112}{1247}$, $\frac{1}{2682281336307046371117292052866419848935586149527745927057} a^{20} - \frac{39674220172996466345106966863208389147011460109963470}{2682281336307046371117292052866419848935586149527745927057} a^{19} - \frac{80746899780626243980437092638920860484720375498766477}{2682281336307046371117292052866419848935586149527745927057} a^{18} - \frac{1244766220169038840650692183109900065585106887303137450}{2682281336307046371117292052866419848935586149527745927057} a^{17} + \frac{29253935488159438248799255495524450768746993924531543854}{2682281336307046371117292052866419848935586149527745927057} a^{16} - \frac{1219158506240717852454838353149295045118762809565424680387}{2682281336307046371117292052866419848935586149527745927057} a^{15} + \frac{844202281967213283680567468354079751572413379052467247036}{2682281336307046371117292052866419848935586149527745927057} a^{14} - \frac{44778331570954259948242241580080248556679058066478234136}{206329333562080490085945542528186142225814319194441994389} a^{13} - \frac{384005030891568436117602796912060601320366306640760009036}{2682281336307046371117292052866419848935586149527745927057} a^{12} + \frac{143294366815656543712066656287861777603371710575756249140}{2682281336307046371117292052866419848935586149527745927057} a^{11} - \frac{859454141618027114979983659304880285896595169924925188985}{2682281336307046371117292052866419848935586149527745927057} a^{10} + \frac{796151059363926656800664690733836558962204360893468119523}{2682281336307046371117292052866419848935586149527745927057} a^{9} - \frac{187261937839232793885657866590562815983365335834120669816}{2682281336307046371117292052866419848935586149527745927057} a^{8} - \frac{766365756275636633755729184181069310348810498232012514893}{2682281336307046371117292052866419848935586149527745927057} a^{7} + \frac{958894818691416028702864634116546946736526055886852304667}{2682281336307046371117292052866419848935586149527745927057} a^{6} - \frac{886619650370934255768171708443929556061215701628564001638}{2682281336307046371117292052866419848935586149527745927057} a^{5} - \frac{857681136825798548010082982011745135286222847006353715859}{2682281336307046371117292052866419848935586149527745927057} a^{4} - \frac{913816312634077769244637213571847210831179901067659978902}{2682281336307046371117292052866419848935586149527745927057} a^{3} - \frac{395765717918982717784363925974714313093303772684778450134}{2682281336307046371117292052866419848935586149527745927057} a^{2} + \frac{1124416419045466823667031598905463676637173029142088645500}{2682281336307046371117292052866419848935586149527745927057} a - \frac{40607562087130944406733873311984931304031418065930288476}{206329333562080490085945542528186142225814319194441994389}$

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

Class group and class number

$C_{7}\times C_{7}$, which has order $49$ (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:  $11$
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:  \( 42748360893.35298 \) (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

\(\Q(\zeta_{7})^+\), 7.1.204024399103.1

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 $21$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $21$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$