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Properties

Label 21.3.36374218133...6207.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 251^{18}$
Root discriminant $550.80$
Ramified primes $7, 251$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (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![7016855542291, 1165451658700, 6818751637159, -3552798817939, 131571710354, -270952034724, 136223349179, -9335801376, 6624085180, -4992150763, 1629066924, -358447607, 43252952, -2299843, 228637, -26511, 8277, -1728, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 - 1728*x^17 + 8277*x^16 - 26511*x^15 + 228637*x^14 - 2299843*x^13 + 43252952*x^12 - 358447607*x^11 + 1629066924*x^10 - 4992150763*x^9 + 6624085180*x^8 - 9335801376*x^7 + 136223349179*x^6 - 270952034724*x^5 + 131571710354*x^4 - 3552798817939*x^3 + 6818751637159*x^2 + 1165451658700*x + 7016855542291)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 - 1728*x^17 + 8277*x^16 - 26511*x^15 + 228637*x^14 - 2299843*x^13 + 43252952*x^12 - 358447607*x^11 + 1629066924*x^10 - 4992150763*x^9 + 6624085180*x^8 - 9335801376*x^7 + 136223349179*x^6 - 270952034724*x^5 + 131571710354*x^4 - 3552798817939*x^3 + 6818751637159*x^2 + 1165451658700*x + 7016855542291, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} - 1728 x^{17} + 8277 x^{16} - 26511 x^{15} + 228637 x^{14} - 2299843 x^{13} + 43252952 x^{12} - 358447607 x^{11} + 1629066924 x^{10} - 4992150763 x^{9} + 6624085180 x^{8} - 9335801376 x^{7} + 136223349179 x^{6} - 270952034724 x^{5} + 131571710354 x^{4} - 3552798817939 x^{3} + 6818751637159 x^{2} + 1165451658700 x + 7016855542291 \)

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:  \(-3637421813338787913991133669958840202605967577382248356207=-\,7^{17}\cdot 251^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $550.80$
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:  $7, 251$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{3}{8} a^{5} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{4016} a^{18} + \frac{43}{4016} a^{17} + \frac{125}{2008} a^{16} + \frac{39}{2008} a^{15} - \frac{459}{4016} a^{14} - \frac{41}{4016} a^{13} - \frac{17}{251} a^{12} - \frac{157}{4016} a^{11} - \frac{26}{251} a^{10} + \frac{419}{4016} a^{9} + \frac{57}{502} a^{8} - \frac{937}{4016} a^{7} - \frac{249}{1004} a^{6} - \frac{1421}{4016} a^{5} - \frac{361}{4016} a^{4} - \frac{183}{502} a^{3} - \frac{873}{2008} a^{2} - \frac{1675}{4016} a - \frac{409}{4016}$, $\frac{1}{791152} a^{19} - \frac{17}{395576} a^{18} - \frac{13603}{791152} a^{17} + \frac{2253}{49447} a^{16} + \frac{41727}{791152} a^{15} - \frac{4603}{197788} a^{14} - \frac{23219}{791152} a^{13} - \frac{91159}{791152} a^{12} + \frac{118097}{791152} a^{11} - \frac{45359}{791152} a^{10} - \frac{17751}{791152} a^{9} - \frac{39563}{791152} a^{8} - \frac{183863}{791152} a^{7} + \frac{148061}{791152} a^{6} + \frac{21625}{49447} a^{5} + \frac{20811}{791152} a^{4} + \frac{95641}{197788} a^{3} - \frac{10303}{791152} a^{2} + \frac{113479}{395576} a - \frac{173323}{791152}$, $\frac{1}{87264262264273929133595191473039871309853729892133128446537520340361560311743780071510365857319024343602279854557168} a^{20} - \frac{8916309727168894870393262356898979021980912469078774672440839427790537682499794212140426224409294688886128317}{43632131132136964566797595736519935654926864946066564223268760170180780155871890035755182928659512171801139927278584} a^{19} - \frac{3448488871537387827528705618985686870849973113160348511697768259487984032888654790104574908103605551693951330595}{43632131132136964566797595736519935654926864946066564223268760170180780155871890035755182928659512171801139927278584} a^{18} + \frac{3800642715347227919925614783575776857120122994688484647638293209443790867231515627828590319540000328513775919453809}{87264262264273929133595191473039871309853729892133128446537520340361560311743780071510365857319024343602279854557168} a^{17} - \frac{1849361700484948645835694462434328872828092188737829441863689168715774657406845867072144348919997378349865824665531}{87264262264273929133595191473039871309853729892133128446537520340361560311743780071510365857319024343602279854557168} a^{16} - \frac{220566380455693184889733112821616187808375301836450266220374490723849920127317322424724298781797207013643695118585}{10908032783034241141699398934129983913731716236516641055817190042545195038967972508938795732164878042950284981819646} a^{15} + \frac{490464709081041264552963780969541898701531657926949901200423975514143427334236147684736311974002519508711874933366}{5454016391517120570849699467064991956865858118258320527908595021272597519483986254469397866082439021475142490909823} a^{14} + \frac{73616914270383007038640955641296780828316954671947582248080355738573811683509460326955605610830290683856009306949}{10908032783034241141699398934129983913731716236516641055817190042545195038967972508938795732164878042950284981819646} a^{13} + \frac{9934182609704483180290864928746310359775146801353637324782888353156441071368871338776248422832086158255685138391385}{87264262264273929133595191473039871309853729892133128446537520340361560311743780071510365857319024343602279854557168} a^{12} - \frac{2327426542866737321990465674282277246047758225190399420046465627599465529837696901483781354379924592337965576578497}{21816065566068482283398797868259967827463432473033282111634380085090390077935945017877591464329756085900569963639292} a^{11} + \frac{18273684463364868602099331544675023322073442682262773329055023204241245552506248823137805529288669401750359211887889}{87264262264273929133595191473039871309853729892133128446537520340361560311743780071510365857319024343602279854557168} a^{10} - \frac{5208384534284799909320737848413848139381593723416361295474408038986141047075312069233865103902467981579815921797}{35531051410534987432245599133973888969810150607546062071065765610896400778397304589377184795325335644789201895178} a^{9} - \frac{16167251768223527676669677808954202354623653953784184961849023122549852747847675321152652316303079289076572197064039}{87264262264273929133595191473039871309853729892133128446537520340361560311743780071510365857319024343602279854557168} a^{8} + \frac{6519171446320188933094490384770477343413056967428427842138201060791977961606755114904157588399278272332705896339}{40930704626770135616132829021125643203496120962538990828582326613678030164983011290577094679793163388181181920524} a^{7} + \frac{1879724916510507399917170976776158949902506692763235345155259182223601787051640454031409385750311052729464859000949}{21816065566068482283398797868259967827463432473033282111634380085090390077935945017877591464329756085900569963639292} a^{6} + \frac{3300087970280858064362639112658356721378384218348982457560211135459790882372135528224949915042719262158053219903959}{43632131132136964566797595736519935654926864946066564223268760170180780155871890035755182928659512171801139927278584} a^{5} + \frac{4231081449052359345461657620695260543204290278730095907409087993659211911906303425201439072716077351805881805674255}{87264262264273929133595191473039871309853729892133128446537520340361560311743780071510365857319024343602279854557168} a^{4} - \frac{1331773054124928645303549099222965383959112041448503793673830617417409138578160478336175637495297014844668932373745}{6712635558790302241045783959464605485373363837856394495887501564643196947057213851654643527486078795661713834965936} a^{3} + \frac{1003760221727935617644242161766808585847298355327665998529002203215199835497604411736769745045956993951898234828417}{10908032783034241141699398934129983913731716236516641055817190042545195038967972508938795732164878042950284981819646} a^{2} + \frac{7880411459310900834589769199401546131877348065722374526150041648270088668554174843922435987631307652494143541102899}{21816065566068482283398797868259967827463432473033282111634380085090390077935945017877591464329756085900569963639292} a + \frac{10146208355841045668007088058342733111523301840320757080389350356355279579223049143730992751295209166670654431426425}{87264262264273929133595191473039871309853729892133128446537520340361560311743780071510365857319024343602279854557168}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 136686531300247910000 \) (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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$
251Data not computed