Properties

Label 18.0.414...208.1
Degree $18$
Signature $[0, 9]$
Discriminant $-4.145\times 10^{22}$
Root discriminant \(18.05\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4)
 
gp: K = bnfinit(y^18 - 3*y^17 - 6*y^16 + 27*y^15 + 18*y^14 - 129*y^13 - 3*y^12 + 354*y^11 - 162*y^10 - 464*y^9 + 426*y^8 + 150*y^7 - 390*y^6 + 252*y^5 + 84*y^4 - 216*y^3 + 132*y^2 - 36*y + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4)
 

\( x^{18} - 3 x^{17} - 6 x^{16} + 27 x^{15} + 18 x^{14} - 129 x^{13} - 3 x^{12} + 354 x^{11} - 162 x^{10} - 464 x^{9} + 426 x^{8} + 150 x^{7} - 390 x^{6} + 252 x^{5} + 84 x^{4} - 216 x^{3} + \cdots + 4 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-41451359947637504606208\) \(\medspace = -\,2^{26}\cdot 3^{31}\) 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:  \(18.05\)
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:  $2^{11/6}3^{97/54}\approx 25.64126624149289$
Ramified primes:   \(2\), \(3\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}$, $\frac{1}{14}a^{15}-\frac{1}{14}a^{14}+\frac{1}{7}a^{13}-\frac{1}{7}a^{12}+\frac{1}{14}a^{11}+\frac{3}{14}a^{10}-\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{1}{14}a^{7}+\frac{3}{14}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{3}+\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{238}a^{16}-\frac{3}{119}a^{15}-\frac{3}{17}a^{14}-\frac{33}{238}a^{13}+\frac{39}{238}a^{12}-\frac{79}{238}a^{11}-\frac{19}{238}a^{10}-\frac{117}{238}a^{9}+\frac{3}{17}a^{8}-\frac{43}{119}a^{7}+\frac{99}{238}a^{6}+\frac{30}{119}a^{5}-\frac{2}{7}a^{4}+\frac{2}{119}a^{3}-\frac{32}{119}a^{2}-\frac{5}{17}a-\frac{5}{119}$, $\frac{1}{9685173675758}a^{17}+\frac{2093318618}{4842586837879}a^{16}-\frac{155228015973}{9685173675758}a^{15}-\frac{2383119769407}{9685173675758}a^{14}+\frac{123474857847}{1383596239394}a^{13}+\frac{759973958645}{4842586837879}a^{12}+\frac{4182205203745}{9685173675758}a^{11}-\frac{2489213685323}{9685173675758}a^{10}+\frac{4676277163443}{9685173675758}a^{9}+\frac{1132457339017}{4842586837879}a^{8}-\frac{2842683667697}{9685173675758}a^{7}-\frac{1038359220419}{9685173675758}a^{6}+\frac{1028802196747}{4842586837879}a^{5}-\frac{1914193135891}{4842586837879}a^{4}+\frac{37584832387}{284858049287}a^{3}+\frac{84528298685}{4842586837879}a^{2}-\frac{1525991999611}{4842586837879}a-\frac{667678476036}{4842586837879}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Trivial group, which has order $1$

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:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{112159473}{390169346} a^{17} + \frac{298357011}{390169346} a^{16} + \frac{773170831}{390169346} a^{15} - \frac{1383174396}{195084673} a^{14} - \frac{2935100103}{390169346} a^{13} + \frac{960745367}{27869239} a^{12} + \frac{2383850487}{195084673} a^{11} - \frac{18962744301}{195084673} a^{10} + \frac{2896017147}{195084673} a^{9} + \frac{53298799815}{390169346} a^{8} - \frac{2183409615}{27869239} a^{7} - \frac{25591216657}{390169346} a^{6} + \frac{2522530224}{27869239} a^{5} - \frac{8851488483}{195084673} a^{4} - \frac{426128644}{11475569} a^{3} + \frac{1379281755}{27869239} a^{2} - \frac{4665195144}{195084673} a + \frac{881662172}{195084673} \)  (order $6$) 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:   $\frac{4983793052212}{4842586837879}a^{17}-\frac{26519556670727}{9685173675758}a^{16}-\frac{2030235166020}{284858049287}a^{15}+\frac{246364831492071}{9685173675758}a^{14}+\frac{264312283344625}{9685173675758}a^{13}-\frac{12\!\cdots\!55}{9685173675758}a^{12}-\frac{63126314585035}{1383596239394}a^{11}+\frac{243272905061920}{691798119697}a^{10}-\frac{456419247702139}{9685173675758}a^{9}-\frac{24\!\cdots\!46}{4842586837879}a^{8}+\frac{13\!\cdots\!02}{4842586837879}a^{7}+\frac{12\!\cdots\!90}{4842586837879}a^{6}-\frac{15\!\cdots\!60}{4842586837879}a^{5}+\frac{706583435198161}{4842586837879}a^{4}+\frac{689541134467820}{4842586837879}a^{3}-\frac{854821036261648}{4842586837879}a^{2}+\frac{350920037376086}{4842586837879}a-\frac{6204286001285}{691798119697}$, $\frac{5882778895471}{9685173675758}a^{17}-\frac{15978020207809}{9685173675758}a^{16}-\frac{39949922142661}{9685173675758}a^{15}+\frac{147770957069735}{9685173675758}a^{14}+\frac{148742927981917}{9685173675758}a^{13}-\frac{359706250761495}{4842586837879}a^{12}-\frac{112851113703589}{4842586837879}a^{11}+\frac{20\!\cdots\!79}{9685173675758}a^{10}-\frac{26247862224754}{691798119697}a^{9}-\frac{28\!\cdots\!97}{9685173675758}a^{8}+\frac{49572953779404}{284858049287}a^{7}+\frac{710871837864061}{4842586837879}a^{6}-\frac{952443190982774}{4842586837879}a^{5}+\frac{64892921704763}{691798119697}a^{4}+\frac{387522565607621}{4842586837879}a^{3}-\frac{534351463632341}{4842586837879}a^{2}+\frac{222078265888106}{4842586837879}a-\frac{35491594119355}{4842586837879}$, $\frac{2251776782067}{9685173675758}a^{17}-\frac{241253965259}{569716098574}a^{16}-\frac{20032363977735}{9685173675758}a^{15}+\frac{20502707151028}{4842586837879}a^{14}+\frac{102159644544947}{9685173675758}a^{13}-\frac{206858054500247}{9685173675758}a^{12}-\frac{18221747245581}{569716098574}a^{11}+\frac{306619885436719}{4842586837879}a^{10}+\frac{71963959466041}{1383596239394}a^{9}-\frac{973448776888475}{9685173675758}a^{8}-\frac{311210934593981}{9685173675758}a^{7}+\frac{369989891617550}{4842586837879}a^{6}-\frac{75003332451965}{4842586837879}a^{5}-\frac{32471394472884}{4842586837879}a^{4}+\frac{214007562051221}{4842586837879}a^{3}-\frac{44053426264807}{4842586837879}a^{2}-\frac{30031835203190}{4842586837879}a+\frac{12511127241345}{4842586837879}$, $\frac{1853731809712}{4842586837879}a^{17}-\frac{675553079421}{691798119697}a^{16}-\frac{26810000805757}{9685173675758}a^{15}+\frac{88709006141811}{9685173675758}a^{14}+\frac{54671373994526}{4842586837879}a^{13}-\frac{435826209654265}{9685173675758}a^{12}-\frac{110092733283444}{4842586837879}a^{11}+\frac{12\!\cdots\!87}{9685173675758}a^{10}-\frac{3170022593337}{9685173675758}a^{9}-\frac{912644987744447}{4842586837879}a^{8}+\frac{356449119797224}{4842586837879}a^{7}+\frac{521703114579034}{4842586837879}a^{6}-\frac{28913871019556}{284858049287}a^{5}+\frac{188010193687300}{4842586837879}a^{4}+\frac{274512535790600}{4842586837879}a^{3}-\frac{285171945436519}{4842586837879}a^{2}+\frac{83867904499065}{4842586837879}a-\frac{512749223373}{284858049287}$, $\frac{307229777049}{4842586837879}a^{17}-\frac{4087141389973}{9685173675758}a^{16}+\frac{795553273447}{4842586837879}a^{15}+\frac{33686722803765}{9685173675758}a^{14}-\frac{5563502468331}{1383596239394}a^{13}-\frac{153729314790101}{9685173675758}a^{12}+\frac{244182584304157}{9685173675758}a^{11}+\frac{11504894481307}{284858049287}a^{10}-\frac{805571340878527}{9685173675758}a^{9}-\frac{198660935488426}{4842586837879}a^{8}+\frac{650864893334658}{4842586837879}a^{7}-\frac{87838668240164}{4842586837879}a^{6}-\frac{430759644264860}{4842586837879}a^{5}+\frac{46649967639232}{691798119697}a^{4}-\frac{58550613258457}{4842586837879}a^{3}-\frac{244462729209141}{4842586837879}a^{2}+\frac{175539419014900}{4842586837879}a-\frac{50591779160635}{4842586837879}$, $\frac{9715294164679}{9685173675758}a^{17}-\frac{1742800503214}{691798119697}a^{16}-\frac{35502964957860}{4842586837879}a^{15}+\frac{32808571736401}{1383596239394}a^{14}+\frac{292335197896933}{9685173675758}a^{13}-\frac{563942630002903}{4842586837879}a^{12}-\frac{301106933686229}{4842586837879}a^{11}+\frac{32\!\cdots\!85}{9685173675758}a^{10}+\frac{49090741229753}{9685173675758}a^{9}-\frac{23\!\cdots\!51}{4842586837879}a^{8}+\frac{908240096230406}{4842586837879}a^{7}+\frac{26\!\cdots\!51}{9685173675758}a^{6}-\frac{12\!\cdots\!81}{4842586837879}a^{5}+\frac{513968294976078}{4842586837879}a^{4}+\frac{715080720125038}{4842586837879}a^{3}-\frac{723387451197230}{4842586837879}a^{2}+\frac{241101652563476}{4842586837879}a-\frac{4339968489950}{691798119697}$, $\frac{1079482508886}{4842586837879}a^{17}-\frac{420547107586}{691798119697}a^{16}-\frac{14586904117017}{9685173675758}a^{15}+\frac{54448611167815}{9685173675758}a^{14}+\frac{26828116886674}{4842586837879}a^{13}-\frac{132374547755475}{4842586837879}a^{12}-\frac{11070193628099}{1383596239394}a^{11}+\frac{746130571046923}{9685173675758}a^{10}-\frac{10933504726333}{691798119697}a^{9}-\frac{74762335054469}{691798119697}a^{8}+\frac{652820861823491}{9685173675758}a^{7}+\frac{28760995637253}{569716098574}a^{6}-\frac{361617821165679}{4842586837879}a^{5}+\frac{191364401688764}{4842586837879}a^{4}+\frac{135187896259190}{4842586837879}a^{3}-\frac{208995524799245}{4842586837879}a^{2}+\frac{95208336294538}{4842586837879}a-\frac{15644641880655}{4842586837879}$, $\frac{1079482508886}{4842586837879}a^{17}-\frac{420547107586}{691798119697}a^{16}-\frac{14586904117017}{9685173675758}a^{15}+\frac{54448611167815}{9685173675758}a^{14}+\frac{26828116886674}{4842586837879}a^{13}-\frac{132374547755475}{4842586837879}a^{12}-\frac{11070193628099}{1383596239394}a^{11}+\frac{746130571046923}{9685173675758}a^{10}-\frac{10933504726333}{691798119697}a^{9}-\frac{74762335054469}{691798119697}a^{8}+\frac{652820861823491}{9685173675758}a^{7}+\frac{28760995637253}{569716098574}a^{6}-\frac{361617821165679}{4842586837879}a^{5}+\frac{191364401688764}{4842586837879}a^{4}+\frac{135187896259190}{4842586837879}a^{3}-\frac{208995524799245}{4842586837879}a^{2}+\frac{95208336294538}{4842586837879}a-\frac{10802055042776}{4842586837879}$ Copy content Toggle raw display
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:  \( 42016.82501482997 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 42016.82501482997 \cdot 1}{6\cdot\sqrt{41451359947637504606208}}\cr\approx \mathstrut & 0.524954188320811 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4)
 
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^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4, 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^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4);
 
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^18 - 3*x^17 - 6*x^16 + 27*x^15 + 18*x^14 - 129*x^13 - 3*x^12 + 354*x^11 - 162*x^10 - 464*x^9 + 426*x^8 + 150*x^7 - 390*x^6 + 252*x^5 + 84*x^4 - 216*x^3 + 132*x^2 - 36*x + 4);
 
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

$C_3\times S_3^2$ (as 18T46):

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 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.216.1, 6.0.139968.1, 6.0.314928.1

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 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.0.1624959306694656.38

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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
\(2\) Copy content Toggle raw display 2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.22.27$x^{12} + 8 x^{9} + 16 x^{8} + 4 x^{7} + 2 x^{6} + 40 x^{5} + 104 x^{4} - 48 x^{3} - 12 x^{2} + 56 x + 196$$6$$2$$22$$C_6\times S_3$$[3]_{3}^{6}$
\(3\) Copy content Toggle raw display Deg $18$$18$$1$$31$