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Properties

Label 21.3.928...699.1
Degree $21$
Signature $[3, 9]$
Discriminant $-9.284\times 10^{24}$
Root discriminant \(15.45\)
Ramified primes $11,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_7$ (as 21T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 5*x^20 + 11*x^19 - 13*x^18 + 7*x^17 + 4*x^16 - 16*x^15 + 13*x^14 - 6*x^13 + 2*x^12 - 2*x^11 + 21*x^10 - 28*x^8 + 29*x^7 - 16*x^6 - 18*x^5 + 17*x^4 - 9*x^3 - 2*x^2 + 3*x - 1)
 
gp: K = bnfinit(y^21 - 5*y^20 + 11*y^19 - 13*y^18 + 7*y^17 + 4*y^16 - 16*y^15 + 13*y^14 - 6*y^13 + 2*y^12 - 2*y^11 + 21*y^10 - 28*y^8 + 29*y^7 - 16*y^6 - 18*y^5 + 17*y^4 - 9*y^3 - 2*y^2 + 3*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 5*x^20 + 11*x^19 - 13*x^18 + 7*x^17 + 4*x^16 - 16*x^15 + 13*x^14 - 6*x^13 + 2*x^12 - 2*x^11 + 21*x^10 - 28*x^8 + 29*x^7 - 16*x^6 - 18*x^5 + 17*x^4 - 9*x^3 - 2*x^2 + 3*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 5*x^20 + 11*x^19 - 13*x^18 + 7*x^17 + 4*x^16 - 16*x^15 + 13*x^14 - 6*x^13 + 2*x^12 - 2*x^11 + 21*x^10 - 28*x^8 + 29*x^7 - 16*x^6 - 18*x^5 + 17*x^4 - 9*x^3 - 2*x^2 + 3*x - 1)
 

\( x^{21} - 5 x^{20} + 11 x^{19} - 13 x^{18} + 7 x^{17} + 4 x^{16} - 16 x^{15} + 13 x^{14} - 6 x^{13} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-9284127557257563917891699\) \(\medspace = -\,11^{9}\cdot 13^{14}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(15.45\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $11^{1/2}13^{2/3}\approx 18.336871607339905$
Ramified primes:   \(11\), \(13\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-11}) \)
$\card{ \Aut(K/\Q) }$:  $3$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{6}a^{19}-\frac{1}{6}a^{18}-\frac{1}{6}a^{15}+\frac{1}{3}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}+\frac{1}{6}a^{11}-\frac{1}{2}a^{10}+\frac{1}{6}a^{9}+\frac{1}{3}a^{8}+\frac{1}{6}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{34943356356}a^{20}-\frac{478427243}{8735839089}a^{19}-\frac{5663418545}{34943356356}a^{18}+\frac{346838965}{5823892726}a^{17}-\frac{5211645547}{34943356356}a^{16}-\frac{96384023}{34943356356}a^{15}+\frac{16530543305}{34943356356}a^{14}-\frac{7114510229}{17471678178}a^{13}-\frac{1100485849}{8735839089}a^{12}-\frac{719597925}{5823892726}a^{11}-\frac{628934344}{8735839089}a^{10}+\frac{2915129115}{11647785452}a^{9}+\frac{1838369019}{11647785452}a^{8}-\frac{13439698727}{34943356356}a^{7}+\frac{8455952215}{17471678178}a^{6}+\frac{443829009}{5823892726}a^{5}+\frac{4252675262}{8735839089}a^{4}-\frac{11116417579}{34943356356}a^{3}+\frac{4181463053}{8735839089}a^{2}-\frac{3979672439}{17471678178}a-\frac{9477342083}{34943356356}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{1373777741}{17471678178}a^{20}-\frac{249749828}{2911946363}a^{19}-\frac{12747502901}{17471678178}a^{18}+\frac{23106040973}{8735839089}a^{17}-\frac{24277330875}{5823892726}a^{16}+\frac{21370957203}{5823892726}a^{15}-\frac{25911466825}{17471678178}a^{14}-\frac{23415256141}{8735839089}a^{13}+\frac{27946490911}{8735839089}a^{12}-\frac{5948945444}{2911946363}a^{11}+\frac{5279732984}{2911946363}a^{10}-\frac{1353336699}{5823892726}a^{9}+\frac{137473699219}{17471678178}a^{8}-\frac{70607087615}{17471678178}a^{7}-\frac{49172880854}{8735839089}a^{6}+\frac{61041476687}{8735839089}a^{5}-\frac{64866351470}{8735839089}a^{4}-\frac{15308126693}{5823892726}a^{3}+\frac{24379509319}{8735839089}a^{2}-\frac{5369749938}{2911946363}a-\frac{1988879987}{17471678178}$, $a$, $\frac{1507176787}{2911946363}a^{20}-\frac{47661401813}{17471678178}a^{19}+\frac{105904393417}{17471678178}a^{18}-\frac{57503939546}{8735839089}a^{17}+\frac{15296027612}{8735839089}a^{16}+\frac{30889489467}{5823892726}a^{15}-\frac{31815956165}{2911946363}a^{14}+\frac{45201661963}{5823892726}a^{13}-\frac{2686865061}{5823892726}a^{12}-\frac{23024836927}{17471678178}a^{11}-\frac{10148808673}{17471678178}a^{10}+\frac{183549976703}{17471678178}a^{9}-\frac{4107248921}{2911946363}a^{8}-\frac{125257191955}{5823892726}a^{7}+\frac{170264271625}{8735839089}a^{6}-\frac{10255321333}{8735839089}a^{5}-\frac{45015675904}{2911946363}a^{4}+\frac{38872376996}{2911946363}a^{3}-\frac{30618886355}{17471678178}a^{2}-\frac{74880142325}{17471678178}a+\frac{45992271689}{17471678178}$, $\frac{15964418909}{34943356356}a^{20}-\frac{37704663113}{17471678178}a^{19}+\frac{160969440469}{34943356356}a^{18}-\frac{94394807047}{17471678178}a^{17}+\frac{100304050853}{34943356356}a^{16}+\frac{70791087071}{34943356356}a^{15}-\frac{86452743401}{11647785452}a^{14}+\frac{48929638393}{8735839089}a^{13}-\frac{67411523929}{17471678178}a^{12}+\frac{2380859284}{2911946363}a^{11}-\frac{17507071999}{17471678178}a^{10}+\frac{355848794507}{34943356356}a^{9}+\frac{98666192197}{34943356356}a^{8}-\frac{104209103303}{11647785452}a^{7}+\frac{247490576891}{17471678178}a^{6}-\frac{166354224907}{17471678178}a^{5}-\frac{95502116122}{8735839089}a^{4}+\frac{43146773595}{11647785452}a^{3}-\frac{34267308655}{5823892726}a^{2}-\frac{2275650606}{2911946363}a+\frac{25891844791}{34943356356}$, $\frac{33417054839}{34943356356}a^{20}-\frac{37519834267}{8735839089}a^{19}+\frac{94936142767}{11647785452}a^{18}-\frac{43389203853}{5823892726}a^{17}+\frac{14974247009}{11647785452}a^{16}+\frac{214184473655}{34943356356}a^{15}-\frac{152393721555}{11647785452}a^{14}+\frac{102774257041}{17471678178}a^{13}-\frac{5409353858}{2911946363}a^{12}+\frac{6349753053}{5823892726}a^{11}-\frac{12255749344}{8735839089}a^{10}+\frac{637067831587}{34943356356}a^{9}+\frac{380511928303}{34943356356}a^{8}-\frac{920691105241}{34943356356}a^{7}+\frac{239744065277}{17471678178}a^{6}-\frac{27275180827}{5823892726}a^{5}-\frac{173670644012}{8735839089}a^{4}+\frac{232499536475}{34943356356}a^{3}-\frac{26657203720}{8735839089}a^{2}-\frac{43499755175}{17471678178}a+\frac{21655573727}{34943356356}$, $\frac{5623858069}{34943356356}a^{20}-\frac{1676712898}{2911946363}a^{19}+\frac{12338632545}{11647785452}a^{18}-\frac{25278857683}{17471678178}a^{17}+\frac{55571284109}{34943356356}a^{16}-\frac{32429012395}{34943356356}a^{15}-\frac{23822605031}{34943356356}a^{14}+\frac{1216084373}{5823892726}a^{13}-\frac{8621528547}{2911946363}a^{12}+\frac{16069956589}{17471678178}a^{11}-\frac{11272059967}{8735839089}a^{10}+\frac{146958204377}{34943356356}a^{9}+\frac{47211844807}{11647785452}a^{8}+\frac{146638604545}{34943356356}a^{7}+\frac{63443915549}{17471678178}a^{6}-\frac{27346788565}{5823892726}a^{5}-\frac{25669250344}{8735839089}a^{4}-\frac{199784753999}{34943356356}a^{3}-\frac{13357788488}{2911946363}a^{2}-\frac{16397762405}{17471678178}a-\frac{71004623275}{34943356356}$, $\frac{28520593511}{17471678178}a^{20}-\frac{133559906833}{17471678178}a^{19}+\frac{133032563404}{8735839089}a^{18}-\frac{132431661113}{8735839089}a^{17}+\frac{84636275485}{17471678178}a^{16}+\frac{24112404878}{2911946363}a^{15}-\frac{363661027775}{17471678178}a^{14}+\frac{179871978653}{17471678178}a^{13}-\frac{11839525059}{5823892726}a^{12}+\frac{48980591839}{17471678178}a^{11}-\frac{80066023949}{17471678178}a^{10}+\frac{294391827529}{8735839089}a^{9}+\frac{183549976703}{17471678178}a^{8}-\frac{411610055917}{8735839089}a^{7}+\frac{225662817977}{8735839089}a^{6}-\frac{19300158821}{2911946363}a^{5}-\frac{266940662932}{8735839089}a^{4}+\frac{214756034263}{17471678178}a^{3}-\frac{7817026541}{5823892726}a^{2}-\frac{29220024459}{5823892726}a+\frac{5340819104}{8735839089}$, $\frac{19753492609}{34943356356}a^{20}-\frac{21444043216}{8735839089}a^{19}+\frac{148591483987}{34943356356}a^{18}-\frac{16095413159}{5823892726}a^{17}-\frac{21715910317}{11647785452}a^{16}+\frac{205717455977}{34943356356}a^{15}-\frac{280696132475}{34943356356}a^{14}+\frac{20964529087}{17471678178}a^{13}+\frac{6920864357}{2911946363}a^{12}-\frac{8929372171}{5823892726}a^{11}-\frac{333381056}{8735839089}a^{10}+\frac{372391908109}{34943356356}a^{9}+\frac{96644609563}{11647785452}a^{8}-\frac{630820635007}{34943356356}a^{7}+\frac{29055044841}{5823892726}a^{6}+\frac{25165052967}{5823892726}a^{5}-\frac{148763950417}{8735839089}a^{4}+\frac{102275918513}{34943356356}a^{3}+\frac{19338159151}{8735839089}a^{2}-\frac{54044616845}{17471678178}a+\frac{29439294865}{34943356356}$, $\frac{1149880739}{17471678178}a^{20}-\frac{42218338}{8735839089}a^{19}-\frac{15767625629}{17471678178}a^{18}+\frac{24571472999}{8735839089}a^{17}-\frac{24895587753}{5823892726}a^{16}+\frac{22338916163}{5823892726}a^{15}-\frac{28864954147}{17471678178}a^{14}-\frac{7900914253}{2911946363}a^{13}+\frac{29551048078}{8735839089}a^{12}-\frac{26458429751}{8735839089}a^{11}+\frac{7451528198}{2911946363}a^{10}-\frac{4294185569}{5823892726}a^{9}+\frac{141190855903}{17471678178}a^{8}-\frac{52665994207}{17471678178}a^{7}-\frac{42551265308}{8735839089}a^{6}+\frac{18640738239}{2911946363}a^{5}-\frac{77625008024}{8735839089}a^{4}-\frac{16023996365}{17471678178}a^{3}+\frac{13339427257}{8735839089}a^{2}-\frac{17388475148}{8735839089}a+\frac{23903165323}{17471678178}$, $\frac{12867859582}{8735839089}a^{20}-\frac{61128068345}{8735839089}a^{19}+\frac{124149125563}{8735839089}a^{18}-\frac{129094704995}{8735839089}a^{17}+\frac{52004068783}{8735839089}a^{16}+\frac{54041935265}{8735839089}a^{15}-\frac{54972893562}{2911946363}a^{14}+\frac{32915593931}{2911946363}a^{13}-\frac{34766533450}{8735839089}a^{12}+\frac{40839254998}{8735839089}a^{11}-\frac{13883271498}{2911946363}a^{10}+\frac{89606774356}{2911946363}a^{9}+\frac{58128685648}{8735839089}a^{8}-\frac{373105058068}{8735839089}a^{7}+\frac{203751113065}{8735839089}a^{6}-\frac{93459084967}{8735839089}a^{5}-\frac{198871703323}{8735839089}a^{4}+\frac{39477662181}{2911946363}a^{3}-\frac{27339456053}{8735839089}a^{2}-\frac{16353309574}{8735839089}a+\frac{12433279112}{8735839089}$, $\frac{11694334139}{17471678178}a^{20}-\frac{27045464426}{8735839089}a^{19}+\frac{36164373039}{5823892726}a^{18}-\frac{58025775695}{8735839089}a^{17}+\frac{19983884023}{5823892726}a^{16}+\frac{25816471723}{17471678178}a^{15}-\frac{135884008181}{17471678178}a^{14}+\frac{48186893479}{8735839089}a^{13}-\frac{12653499997}{2911946363}a^{12}+\frac{37265924332}{8735839089}a^{11}-\frac{8613841881}{2911946363}a^{10}+\frac{244274210585}{17471678178}a^{9}+\frac{73514359519}{17471678178}a^{8}-\frac{277064373629}{17471678178}a^{7}+\frac{74647586567}{8735839089}a^{6}-\frac{72863985466}{8735839089}a^{5}-\frac{48411572435}{8735839089}a^{4}+\frac{60785139917}{17471678178}a^{3}-\frac{11484862868}{2911946363}a^{2}+\frac{5559739372}{2911946363}a-\frac{6229234573}{17471678178}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 10371.33223 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 10371.33223 \cdot 1}{2\cdot\sqrt{9284127557257563917891699}}\cr\approx \mathstrut & 0.2077986892 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 5*x^20 + 11*x^19 - 13*x^18 + 7*x^17 + 4*x^16 - 16*x^15 + 13*x^14 - 6*x^13 + 2*x^12 - 2*x^11 + 21*x^10 - 28*x^8 + 29*x^7 - 16*x^6 - 18*x^5 + 17*x^4 - 9*x^3 - 2*x^2 + 3*x - 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^21 - 5*x^20 + 11*x^19 - 13*x^18 + 7*x^17 + 4*x^16 - 16*x^15 + 13*x^14 - 6*x^13 + 2*x^12 - 2*x^11 + 21*x^10 - 28*x^8 + 29*x^7 - 16*x^6 - 18*x^5 + 17*x^4 - 9*x^3 - 2*x^2 + 3*x - 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - 5*x^20 + 11*x^19 - 13*x^18 + 7*x^17 + 4*x^16 - 16*x^15 + 13*x^14 - 6*x^13 + 2*x^12 - 2*x^11 + 21*x^10 - 28*x^8 + 29*x^7 - 16*x^6 - 18*x^5 + 17*x^4 - 9*x^3 - 2*x^2 + 3*x - 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 5*x^20 + 11*x^19 - 13*x^18 + 7*x^17 + 4*x^16 - 16*x^15 + 13*x^14 - 6*x^13 + 2*x^12 - 2*x^11 + 21*x^10 - 28*x^8 + 29*x^7 - 16*x^6 - 18*x^5 + 17*x^4 - 9*x^3 - 2*x^2 + 3*x - 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$F_7$ (as 21T4):

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

Intermediate fields

3.3.169.1, 7.1.38014691.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: data not computed
Degree 7 sibling: 7.1.38014691.1
Degree 14 sibling: 14.0.15896284050080291.1
Minimal sibling: 7.1.38014691.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{3}{,}\,{\href{/padicField/2.3.0.1}{3} }$ ${\href{/padicField/3.3.0.1}{3} }^{7}$ ${\href{/padicField/5.7.0.1}{7} }^{3}$ ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$ R R ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.3.0.1}{3} }^{7}$ ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.7.0.1}{7} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.7.0.1}{7} }^{3}$ ${\href{/padicField/53.7.0.1}{7} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{7}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(11\) Copy content Toggle raw display 11.3.0.1$x^{3} + 2 x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(13\) Copy content Toggle raw display 13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$