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Properties

Label 21.3.109...167.1
Degree $21$
Signature $[3, 9]$
Discriminant $-1.090\times 10^{25}$
Root discriminant \(15.57\)
Ramified primes $31,67,349$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3\times S_7$ (as 21T74)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 + x^19 - 2*x^18 - 2*x^17 - 3*x^16 - x^15 + 4*x^14 + 15*x^13 + 3*x^12 - 4*x^11 - 29*x^10 - 4*x^9 + 12*x^8 + 23*x^7 + 16*x^6 - 4*x^5 - 7*x^4 - 8*x^3 - x^2 + 1)
 
gp: K = bnfinit(y^21 + y^19 - 2*y^18 - 2*y^17 - 3*y^16 - y^15 + 4*y^14 + 15*y^13 + 3*y^12 - 4*y^11 - 29*y^10 - 4*y^9 + 12*y^8 + 23*y^7 + 16*y^6 - 4*y^5 - 7*y^4 - 8*y^3 - y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 + x^19 - 2*x^18 - 2*x^17 - 3*x^16 - x^15 + 4*x^14 + 15*x^13 + 3*x^12 - 4*x^11 - 29*x^10 - 4*x^9 + 12*x^8 + 23*x^7 + 16*x^6 - 4*x^5 - 7*x^4 - 8*x^3 - x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 + x^19 - 2*x^18 - 2*x^17 - 3*x^16 - x^15 + 4*x^14 + 15*x^13 + 3*x^12 - 4*x^11 - 29*x^10 - 4*x^9 + 12*x^8 + 23*x^7 + 16*x^6 - 4*x^5 - 7*x^4 - 8*x^3 - x^2 + 1)
 

\( x^{21} + x^{19} - 2 x^{18} - 2 x^{17} - 3 x^{16} - x^{15} + 4 x^{14} + 15 x^{13} + 3 x^{12} - 4 x^{11} + \cdots + 1 \) 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:   \(-10904210824444953500612167\) \(\medspace = -\,31^{8}\cdot 67^{3}\cdot 349^{3}\) 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:  \(15.57\)
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:  $31^{1/2}67^{1/2}349^{1/2}\approx 851.3947380621987$
Ramified primes:   \(31\), \(67\), \(349\) 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:  \(\Q(\sqrt{-23383}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4309}a^{19}+\frac{1938}{4309}a^{18}+\frac{1061}{4309}a^{17}+\frac{1473}{4309}a^{16}+\frac{1914}{4309}a^{15}-\frac{2147}{4309}a^{14}-\frac{1353}{4309}a^{13}+\frac{506}{4309}a^{12}+\frac{432}{4309}a^{11}+\frac{540}{4309}a^{10}-\frac{226}{4309}a^{9}+\frac{855}{4309}a^{8}-\frac{783}{4309}a^{7}+\frac{1894}{4309}a^{6}-\frac{1029}{4309}a^{5}+\frac{658}{4309}a^{4}-\frac{996}{4309}a^{3}-\frac{674}{4309}a^{2}+\frac{407}{4309}a+\frac{1535}{4309}$, $\frac{1}{38311319}a^{20}+\frac{2249}{38311319}a^{19}+\frac{9450156}{38311319}a^{18}-\frac{2499569}{38311319}a^{17}-\frac{3672314}{38311319}a^{16}+\frac{11240646}{38311319}a^{15}+\frac{11146208}{38311319}a^{14}+\frac{6849006}{38311319}a^{13}-\frac{2065646}{38311319}a^{12}-\frac{9594830}{38311319}a^{11}+\frac{10655820}{38311319}a^{10}+\frac{989774}{2253607}a^{9}-\frac{6086344}{38311319}a^{8}-\frac{10264353}{38311319}a^{7}-\frac{714924}{2253607}a^{6}-\frac{12535376}{38311319}a^{5}+\frac{6279332}{38311319}a^{4}+\frac{2029357}{38311319}a^{3}-\frac{2118094}{38311319}a^{2}+\frac{14981235}{38311319}a+\frac{4122799}{38311319}$ 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:   $\frac{4195103}{38311319}a^{20}-\frac{14781537}{38311319}a^{19}+\frac{23518699}{38311319}a^{18}-\frac{22749912}{38311319}a^{17}+\frac{28290667}{38311319}a^{16}-\frac{17086035}{38311319}a^{15}-\frac{6399772}{38311319}a^{14}-\frac{3605221}{38311319}a^{13}+\frac{2179763}{38311319}a^{12}-\frac{114962006}{38311319}a^{11}+\frac{226707958}{38311319}a^{10}-\frac{3063631}{2253607}a^{9}+\frac{176642579}{38311319}a^{8}-\frac{409025578}{38311319}a^{7}-\frac{3273509}{2253607}a^{6}+\frac{226923059}{38311319}a^{5}+\frac{75888614}{38311319}a^{4}+\frac{108238953}{38311319}a^{3}-\frac{96237832}{38311319}a^{2}-\frac{33543099}{38311319}a-\frac{40562465}{38311319}$, $\frac{14781537}{38311319}a^{20}-\frac{19323596}{38311319}a^{19}+\frac{14359706}{38311319}a^{18}-\frac{36680873}{38311319}a^{17}+\frac{4500726}{38311319}a^{16}+\frac{2204669}{38311319}a^{15}+\frac{20385633}{38311319}a^{14}+\frac{60746782}{38311319}a^{13}+\frac{127547315}{38311319}a^{12}-\frac{243488370}{38311319}a^{11}-\frac{69576260}{38311319}a^{10}-\frac{11377823}{2253607}a^{9}+\frac{459366814}{38311319}a^{8}+\frac{152137022}{38311319}a^{7}-\frac{9400083}{2253607}a^{6}-\frac{92669026}{38311319}a^{5}-\frac{137604674}{38311319}a^{4}+\frac{62677008}{38311319}a^{3}+\frac{29347996}{38311319}a^{2}+\frac{40562465}{38311319}a+\frac{4195103}{38311319}$, $\frac{573117}{38311319}a^{20}-\frac{25780928}{38311319}a^{19}+\frac{15365737}{38311319}a^{18}-\frac{12567456}{38311319}a^{17}+\frac{58090443}{38311319}a^{16}+\frac{30033579}{38311319}a^{15}+\frac{18405642}{38311319}a^{14}-\frac{34607356}{38311319}a^{13}-\frac{124817870}{38311319}a^{12}-\frac{322319812}{38311319}a^{11}+\frac{205021789}{38311319}a^{10}+\frac{17997010}{2253607}a^{9}+\frac{600642792}{38311319}a^{8}-\frac{430452446}{38311319}a^{7}-\frac{38312866}{2253607}a^{6}-\frac{214806828}{38311319}a^{5}+\frac{192849056}{38311319}a^{4}+\frac{406691412}{38311319}a^{3}+\frac{112895437}{38311319}a^{2}-\frac{23143587}{38311319}a-\frac{90816471}{38311319}$, $\frac{23357077}{38311319}a^{20}-\frac{2842986}{38311319}a^{19}+\frac{28067533}{38311319}a^{18}-\frac{47500986}{38311319}a^{17}-\frac{39615060}{38311319}a^{16}-\frac{68954930}{38311319}a^{15}-\frac{28110885}{38311319}a^{14}+\frac{83350396}{38311319}a^{13}+\frac{333691690}{38311319}a^{12}+\frac{40506764}{38311319}a^{11}-\frac{25899971}{38311319}a^{10}-\frac{37661131}{2253607}a^{9}-\frac{59129329}{38311319}a^{8}+\frac{183558951}{38311319}a^{7}+\frac{28170167}{2253607}a^{6}+\frac{387266363}{38311319}a^{5}-\frac{73484223}{38311319}a^{4}-\frac{129049749}{38311319}a^{3}-\frac{107644921}{38311319}a^{2}-\frac{26652763}{38311319}a-\frac{7539584}{38311319}$, $\frac{3013269}{38311319}a^{20}+\frac{18081601}{38311319}a^{19}-\frac{410216}{1235849}a^{18}+\frac{7904984}{38311319}a^{17}-\frac{56484182}{38311319}a^{16}-\frac{14389906}{38311319}a^{15}-\frac{19066867}{38311319}a^{14}+\frac{47090534}{38311319}a^{13}+\frac{135013368}{38311319}a^{12}+\frac{213956738}{38311319}a^{11}-\frac{211980856}{38311319}a^{10}-\frac{15073791}{2253607}a^{9}-\frac{445484685}{38311319}a^{8}+\frac{480861831}{38311319}a^{7}+\frac{24498704}{2253607}a^{6}+\frac{197709709}{38311319}a^{5}-\frac{1396114}{275621}a^{4}-\frac{325657714}{38311319}a^{3}-\frac{30624013}{38311319}a^{2}-\frac{28808859}{38311319}a+\frac{90566296}{38311319}$, $\frac{16570536}{38311319}a^{20}-\frac{18348847}{38311319}a^{19}+\frac{21848922}{38311319}a^{18}-\frac{40558146}{38311319}a^{17}+\frac{11382073}{38311319}a^{16}-\frac{19776027}{38311319}a^{15}+\frac{14237110}{38311319}a^{14}+\frac{37844124}{38311319}a^{13}+\frac{148977883}{38311319}a^{12}-\frac{216435166}{38311319}a^{11}+\frac{1264249}{38311319}a^{10}-\frac{13492191}{2253607}a^{9}+\frac{477606387}{38311319}a^{8}-\frac{3546392}{38311319}a^{7}-\frac{6625558}{2253607}a^{6}-\frac{126219788}{38311319}a^{5}+\frac{32351889}{38311319}a^{4}+\frac{187613778}{38311319}a^{3}+\frac{34345417}{38311319}a^{2}+\frac{50480732}{38311319}a-\frac{2479581}{1235849}$, $\frac{75410369}{38311319}a^{20}-\frac{16460636}{38311319}a^{19}+\frac{44276101}{38311319}a^{18}-\frac{148348468}{38311319}a^{17}-\frac{137008992}{38311319}a^{16}-\frac{122469658}{38311319}a^{15}+\frac{8566611}{38311319}a^{14}+\frac{344211728}{38311319}a^{13}+\frac{1035620831}{38311319}a^{12}-\frac{172793963}{38311319}a^{11}-\frac{728238402}{38311319}a^{10}-\frac{110562096}{2253607}a^{9}+\frac{484102604}{38311319}a^{8}+\frac{1666412344}{38311319}a^{7}+\frac{60499313}{2253607}a^{6}+\frac{146951161}{38311319}a^{5}-\frac{764770509}{38311319}a^{4}-\frac{331914070}{38311319}a^{3}-\frac{42105256}{38311319}a^{2}+\frac{115278821}{38311319}a-\frac{20234274}{38311319}$, $\frac{18047858}{38311319}a^{20}+\frac{528352}{38311319}a^{19}+\frac{12943704}{38311319}a^{18}-\frac{33950320}{38311319}a^{17}-\frac{33043727}{38311319}a^{16}-\frac{44362016}{38311319}a^{15}-\frac{6058094}{38311319}a^{14}+\frac{68736737}{38311319}a^{13}+\frac{257196688}{38311319}a^{12}+\frac{21083482}{38311319}a^{11}-\frac{145552193}{38311319}a^{10}-\frac{28533397}{2253607}a^{9}+\frac{63149002}{38311319}a^{8}+\frac{352554285}{38311319}a^{7}+\frac{19884280}{2253607}a^{6}+\frac{1152502}{1235849}a^{5}-\frac{111139308}{38311319}a^{4}-\frac{42913758}{38311319}a^{3}-\frac{9533960}{38311319}a^{2}+\frac{1776424}{1235849}a+\frac{27308274}{38311319}$, $\frac{8306894}{38311319}a^{20}+\frac{2587028}{38311319}a^{19}+\frac{13463701}{38311319}a^{18}-\frac{8498072}{38311319}a^{17}-\frac{22765560}{38311319}a^{16}-\frac{34261399}{38311319}a^{15}-\frac{36926310}{38311319}a^{14}+\frac{15920209}{38311319}a^{13}+\frac{129942046}{38311319}a^{12}+\frac{86975320}{38311319}a^{11}+\frac{67436368}{38311319}a^{10}-\frac{11093389}{2253607}a^{9}-\frac{204786595}{38311319}a^{8}-\frac{73634575}{38311319}a^{7}+\frac{8976125}{2253607}a^{6}+\frac{394926085}{38311319}a^{5}+\frac{158821779}{38311319}a^{4}-\frac{51658271}{38311319}a^{3}-\frac{112491494}{38311319}a^{2}-\frac{75434514}{38311319}a+\frac{29445619}{38311319}$, $\frac{1114116}{38311319}a^{20}+\frac{11410199}{38311319}a^{19}-\frac{8394870}{38311319}a^{18}+\frac{9199246}{38311319}a^{17}-\frac{34862000}{38311319}a^{16}-\frac{7506879}{38311319}a^{15}-\frac{15653019}{38311319}a^{14}+\frac{18218880}{38311319}a^{13}+\frac{74315239}{38311319}a^{12}+\frac{134385288}{38311319}a^{11}-\frac{107055736}{38311319}a^{10}-\frac{6064103}{2253607}a^{9}-\frac{9642610}{1235849}a^{8}+\frac{240030244}{38311319}a^{7}+\frac{11559396}{2253607}a^{6}+\frac{124615742}{38311319}a^{5}-\frac{38233529}{38311319}a^{4}-\frac{194374944}{38311319}a^{3}+\frac{36438190}{38311319}a^{2}-\frac{9867489}{38311319}a+\frac{44025926}{38311319}$, $\frac{372047}{1235849}a^{20}-\frac{3219405}{38311319}a^{19}+\frac{19612727}{38311319}a^{18}-\frac{18694036}{38311319}a^{17}-\frac{9025507}{38311319}a^{16}-\frac{36546359}{38311319}a^{15}-\frac{29680789}{38311319}a^{14}+\frac{13054704}{38311319}a^{13}+\frac{134237262}{38311319}a^{12}+\frac{9534749}{38311319}a^{11}+\frac{85967493}{38311319}a^{10}-\frac{11664598}{2253607}a^{9}+\frac{14087363}{38311319}a^{8}-\frac{107473646}{38311319}a^{7}+\frac{1356915}{2253607}a^{6}+\frac{227327236}{38311319}a^{5}+\frac{169609326}{38311319}a^{4}+\frac{167539069}{38311319}a^{3}-\frac{20698799}{38311319}a^{2}-\frac{46309508}{38311319}a-\frac{60354718}{38311319}$ 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:  \( 10492.3673171 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 10492.3673171 \cdot 1}{2\cdot\sqrt{10904210824444953500612167}}\cr\approx \mathstrut & 0.193979203931 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 + x^19 - 2*x^18 - 2*x^17 - 3*x^16 - x^15 + 4*x^14 + 15*x^13 + 3*x^12 - 4*x^11 - 29*x^10 - 4*x^9 + 12*x^8 + 23*x^7 + 16*x^6 - 4*x^5 - 7*x^4 - 8*x^3 - x^2 + 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 + x^19 - 2*x^18 - 2*x^17 - 3*x^16 - x^15 + 4*x^14 + 15*x^13 + 3*x^12 - 4*x^11 - 29*x^10 - 4*x^9 + 12*x^8 + 23*x^7 + 16*x^6 - 4*x^5 - 7*x^4 - 8*x^3 - x^2 + 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 + x^19 - 2*x^18 - 2*x^17 - 3*x^16 - x^15 + 4*x^14 + 15*x^13 + 3*x^12 - 4*x^11 - 29*x^10 - 4*x^9 + 12*x^8 + 23*x^7 + 16*x^6 - 4*x^5 - 7*x^4 - 8*x^3 - x^2 + 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 + x^19 - 2*x^18 - 2*x^17 - 3*x^16 - x^15 + 4*x^14 + 15*x^13 + 3*x^12 - 4*x^11 - 29*x^10 - 4*x^9 + 12*x^8 + 23*x^7 + 16*x^6 - 4*x^5 - 7*x^4 - 8*x^3 - x^2 + 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

$S_3\times S_7$ (as 21T74):

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 non-solvable group of order 30240
The 45 conjugacy class representatives for $S_3\times S_7$
Character table for $S_3\times S_7$

Intermediate fields

3.1.31.1, 7.3.724873.1

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

Degree 42 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $21$ ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.7.0.1}{7} }$ $15{,}\,{\href{/padicField/5.6.0.1}{6} }$ $21$ ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ R ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }$ $15{,}\,{\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{5}$

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
\(31\) Copy content Toggle raw display $\Q_{31}$$x + 28$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.1$x^{2} + 93$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(67\) Copy content Toggle raw display $\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.4.0.1$x^{4} + 8 x^{2} + 54 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} + 8 x^{2} + 54 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} + 8 x^{2} + 54 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(349\) Copy content Toggle raw display $\Q_{349}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{349}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{349}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$