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Properties

Label 21.3.48958786234...5327.1
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 $F_7$ (as 21T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![569387, 1634962, 4310250, -14827477, 2320101, 1287601, 5161290, -3230632, 105237, 43556, 225822, -170639, 42918, -3577, 248, -373, 52, 29, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 52*x^16 - 373*x^15 + 248*x^14 - 3577*x^13 + 42918*x^12 - 170639*x^11 + 225822*x^10 + 43556*x^9 + 105237*x^8 - 3230632*x^7 + 5161290*x^6 + 1287601*x^5 + 2320101*x^4 - 14827477*x^3 + 4310250*x^2 + 1634962*x + 569387)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 29*x^17 + 52*x^16 - 373*x^15 + 248*x^14 - 3577*x^13 + 42918*x^12 - 170639*x^11 + 225822*x^10 + 43556*x^9 + 105237*x^8 - 3230632*x^7 + 5161290*x^6 + 1287601*x^5 + 2320101*x^4 - 14827477*x^3 + 4310250*x^2 + 1634962*x + 569387, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 52 x^{16} - 373 x^{15} + 248 x^{14} - 3577 x^{13} + 42918 x^{12} - 170639 x^{11} + 225822 x^{10} + 43556 x^{9} + 105237 x^{8} - 3230632 x^{7} + 5161290 x^{6} + 1287601 x^{5} + 2320101 x^{4} - 14827477 x^{3} + 4310250 x^{2} + 1634962 x + 569387 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{16} a^{16} - \frac{1}{16} a^{15} + \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{5}{16} a^{6} + \frac{5}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} + \frac{7}{16} a + \frac{5}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{3}{16} a^{7} + \frac{1}{4} a^{5} + \frac{7}{16} a^{4} - \frac{1}{2} a^{3} + \frac{7}{16} a^{2} - \frac{1}{4} a - \frac{3}{16}$, $\frac{1}{1392} a^{18} + \frac{7}{348} a^{17} + \frac{2}{87} a^{16} + \frac{1}{348} a^{15} + \frac{19}{696} a^{14} + \frac{41}{696} a^{13} - \frac{73}{1392} a^{12} - \frac{5}{48} a^{11} + \frac{43}{1392} a^{10} - \frac{53}{348} a^{9} + \frac{239}{1392} a^{8} - \frac{1}{6} a^{7} + \frac{1}{87} a^{6} + \frac{583}{1392} a^{5} + \frac{9}{29} a^{4} - \frac{665}{1392} a^{3} + \frac{10}{87} a^{2} - \frac{379}{1392} a - \frac{17}{87}$, $\frac{1}{5568} a^{19} - \frac{1}{5568} a^{18} + \frac{15}{928} a^{17} - \frac{47}{1856} a^{16} + \frac{3}{1856} a^{15} + \frac{37}{1856} a^{14} - \frac{23}{464} a^{13} + \frac{5}{192} a^{12} - \frac{121}{1856} a^{11} + \frac{1151}{5568} a^{10} - \frac{23}{928} a^{9} + \frac{35}{192} a^{8} + \frac{5}{58} a^{7} - \frac{941}{2784} a^{6} + \frac{257}{696} a^{5} - \frac{67}{174} a^{4} + \frac{1001}{5568} a^{3} + \frac{241}{1856} a^{2} + \frac{161}{464} a - \frac{61}{192}$, $\frac{1}{6043208541674794207816680097453940316162807123120463721862145536} a^{20} - \frac{2426706923062695401090217364713303917656978599307278541105}{503600711806232850651390008121161693013567260260038643488512128} a^{19} + \frac{127797649346013882029795299159943996331492589448521715477371}{2014402847224931402605560032484646772054269041040154573954048512} a^{18} - \frac{164561231335732510059709609427508363767897274835785475553398391}{6043208541674794207816680097453940316162807123120463721862145536} a^{17} + \frac{8463394740474281816974086977747181232914550471610241983698829}{377700533854674637988542506090871269760175445195028982616384096} a^{16} + \frac{6764534421307087575609802622254221191889607746807230945792137}{1510802135418698551954170024363485079040701780780115930465536384} a^{15} + \frac{1052503333952556417989994664923676659862352472257009967455611}{11338102329596236787648555529932345808935848261014003230510592} a^{14} - \frac{722735824570875890792469261684300501713562810897364880374059835}{6043208541674794207816680097453940316162807123120463721862145536} a^{13} + \frac{120896341194537219114642688185615040390057619352849689541623533}{1007201423612465701302780016242323386027134520520077286977024256} a^{12} - \frac{12134262965316319231474644362603311498989155301497479600642951}{188850266927337318994271253045435634880087722597514491308192048} a^{11} - \frac{332378130007159836825234400183748895854021542553291413270283077}{2014402847224931402605560032484646772054269041040154573954048512} a^{10} + \frac{101214370263466164653690379879005724678697102032299909471228045}{464862195513445708293590776727226178166369778701574132450934272} a^{9} + \frac{53953249116629819775699655181862473648674262909909111644115549}{2014402847224931402605560032484646772054269041040154573954048512} a^{8} + \frac{739567722077513043514038288074291070616904031116422506492814625}{3021604270837397103908340048726970158081403561560231860931072768} a^{7} + \frac{334614174394060662208893420850507911636594076042513588523432567}{3021604270837397103908340048726970158081403561560231860931072768} a^{6} + \frac{195210961622644356206889722601494730674903218503179634690537403}{503600711806232850651390008121161693013567260260038643488512128} a^{5} - \frac{983479142183328896723950165606757161179039974065543411342816321}{2014402847224931402605560032484646772054269041040154573954048512} a^{4} - \frac{127995142364284854392684879052587278280589063447405767148235585}{503600711806232850651390008121161693013567260260038643488512128} a^{3} + \frac{1189176411615846044097342386554806249330565766703182272835244487}{6043208541674794207816680097453940316162807123120463721862145536} a^{2} + \frac{647507206202097950372243489997810112512627549773495304420101165}{2014402847224931402605560032484646772054269041040154573954048512} a + \frac{25208522556169884697397696782625070589768982377120960033599947}{464862195513445708293590776727226178166369778701574132450934272}$

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:  \( 72630093220.36017 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

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

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$