Properties

Label 20.0.174...736.1
Degree $20$
Signature $[0, 10]$
Discriminant $1.744\times 10^{24}$
Root discriminant \(16.30\)
Ramified primes $2,67$
Class number $1$
Class group trivial
Galois group $C_2\times A_5$ (as 20T31)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*x + 4)
 
gp: K = bnfinit(y^20 - 4*y^19 + 8*y^18 - 8*y^17 + 5*y^16 + 2*y^15 - 16*y^14 + 30*y^13 - 22*y^12 - 4*y^11 + 34*y^10 - 66*y^9 + 109*y^8 - 136*y^7 + 120*y^6 - 76*y^5 + 48*y^4 - 40*y^3 + 32*y^2 - 16*y + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*x + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*x + 4)
 

\( x^{20} - 4 x^{19} + 8 x^{18} - 8 x^{17} + 5 x^{16} + 2 x^{15} - 16 x^{14} + 30 x^{13} - 22 x^{12} - 4 x^{11} + 34 x^{10} - 66 x^{9} + 109 x^{8} - 136 x^{7} + 120 x^{6} - 76 x^{5} + 48 x^{4} - 40 x^{3} + \cdots + 4 \) Copy content Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1744047395068446282612736\) \(\medspace = 2^{32}\cdot 67^{8}\) 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:  \(16.30\)
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))
 
Ramified primes:   \(2\), \(67\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{4}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{17}-\frac{1}{2}a^{15}-\frac{1}{4}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{159851548}a^{19}-\frac{4703375}{39962887}a^{18}+\frac{6748419}{79925774}a^{17}-\frac{2354475}{159851548}a^{16}-\frac{76917031}{159851548}a^{15}-\frac{5664755}{39962887}a^{14}-\frac{1514615}{79925774}a^{13}+\frac{12706489}{159851548}a^{12}-\frac{11427671}{39962887}a^{11}+\frac{10227295}{79925774}a^{10}+\frac{8360643}{39962887}a^{9}+\frac{1043761}{79925774}a^{8}+\frac{20328221}{159851548}a^{7}+\frac{2022511}{79925774}a^{6}+\frac{198553}{39962887}a^{5}+\frac{19349637}{159851548}a^{4}-\frac{18012177}{39962887}a^{3}-\frac{28244355}{79925774}a^{2}-\frac{9619111}{39962887}a-\frac{20147829}{79925774}$ 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 $1$

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:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{26758413}{159851548} a^{19} - \frac{60872855}{79925774} a^{18} + \frac{122020429}{79925774} a^{17} - \frac{121676233}{79925774} a^{16} + \frac{115506207}{159851548} a^{15} + \frac{15963377}{39962887} a^{14} - \frac{251929171}{79925774} a^{13} + \frac{239526437}{39962887} a^{12} - \frac{159792325}{39962887} a^{11} - \frac{53142461}{39962887} a^{10} + \frac{563142481}{79925774} a^{9} - \frac{993516103}{79925774} a^{8} + \frac{3289992433}{159851548} a^{7} - \frac{1993748639}{79925774} a^{6} + \frac{1672955167}{79925774} a^{5} - \frac{889058877}{79925774} a^{4} + \frac{525398757}{79925774} a^{3} - \frac{216472536}{39962887} a^{2} + \frac{218315452}{39962887} a - \frac{83240710}{39962887} \)  (order $4$) 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:   $\frac{182484689}{159851548}a^{19}-\frac{640373105}{159851548}a^{18}+\frac{261967466}{39962887}a^{17}-\frac{173237178}{39962887}a^{16}+\frac{225400667}{159851548}a^{15}+\frac{572767547}{159851548}a^{14}-\frac{1369422177}{79925774}a^{13}+\frac{1876290125}{79925774}a^{12}-\frac{536220935}{79925774}a^{11}-\frac{590311809}{39962887}a^{10}+\frac{2407820417}{79925774}a^{9}-\frac{2213546281}{39962887}a^{8}+\frac{13720260899}{159851548}a^{7}-\frac{14810265693}{159851548}a^{6}+\frac{2449175043}{39962887}a^{5}-\frac{1225262303}{39962887}a^{4}+\frac{1995272755}{79925774}a^{3}-\frac{2058069531}{79925774}a^{2}+\frac{638830655}{39962887}a-\frac{171717862}{39962887}$, $\frac{65543731}{79925774}a^{19}-\frac{346023687}{159851548}a^{18}+\frac{571865403}{159851548}a^{17}-\frac{151829703}{79925774}a^{16}+\frac{71328043}{39962887}a^{15}+\frac{605781891}{159851548}a^{14}-\frac{1367211941}{159851548}a^{13}+\frac{487411193}{39962887}a^{12}-\frac{103128733}{39962887}a^{11}-\frac{432446027}{79925774}a^{10}+\frac{808605822}{39962887}a^{9}-\frac{2350787063}{79925774}a^{8}+\frac{3855799763}{79925774}a^{7}-\frac{7687497001}{159851548}a^{6}+\frac{5876627707}{159851548}a^{5}-\frac{1351826321}{79925774}a^{4}+\frac{1167655191}{79925774}a^{3}-\frac{1056330925}{79925774}a^{2}+\frac{778589567}{79925774}a-\frac{74202314}{39962887}$, $\frac{394983739}{159851548}a^{19}-\frac{1147849919}{159851548}a^{18}+\frac{1714191703}{159851548}a^{17}-\frac{199386560}{39962887}a^{16}+\frac{454867499}{159851548}a^{15}+\frac{1423003613}{159851548}a^{14}-\frac{4917545993}{159851548}a^{13}+\frac{1438635995}{39962887}a^{12}-\frac{73611069}{39962887}a^{11}-\frac{1962461361}{79925774}a^{10}+\frac{4218649835}{79925774}a^{9}-\frac{3791093360}{39962887}a^{8}+\frac{23195589747}{159851548}a^{7}-\frac{22326057829}{159851548}a^{6}+\frac{13942142083}{159851548}a^{5}-\frac{1855903695}{39962887}a^{4}+\frac{1702390586}{39962887}a^{3}-\frac{3025695325}{79925774}a^{2}+\frac{1651075337}{79925774}a-\frac{189719003}{39962887}$, $\frac{6522125}{159851548}a^{19}+\frac{87664537}{159851548}a^{18}-\frac{224761387}{159851548}a^{17}+\frac{79591307}{39962887}a^{16}-\frac{58019379}{159851548}a^{15}+\frac{153118833}{159851548}a^{14}+\frac{354878841}{159851548}a^{13}-\frac{261530599}{39962887}a^{12}+\frac{243530023}{39962887}a^{11}+\frac{123120469}{79925774}a^{10}-\frac{281281589}{79925774}a^{9}+\frac{430491925}{39962887}a^{8}-\frac{2950622659}{159851548}a^{7}+\frac{4355807191}{159851548}a^{6}-\frac{3596169383}{159851548}a^{5}+\frac{497581053}{39962887}a^{4}-\frac{249629044}{39962887}a^{3}+\frac{509318565}{79925774}a^{2}-\frac{334463291}{79925774}a+\frac{72518701}{39962887}$, $\frac{18044039}{79925774}a^{19}-\frac{100717081}{159851548}a^{18}+\frac{122384591}{159851548}a^{17}+\frac{13447045}{159851548}a^{16}-\frac{25659619}{79925774}a^{15}+\frac{152593327}{159851548}a^{14}-\frac{411317005}{159851548}a^{13}+\frac{417586557}{159851548}a^{12}+\frac{84645890}{39962887}a^{11}-\frac{148593592}{39962887}a^{10}+\frac{318948935}{79925774}a^{9}-\frac{450055345}{79925774}a^{8}+\frac{817015707}{79925774}a^{7}-\frac{1082590025}{159851548}a^{6}+\frac{2233701}{159851548}a^{5}+\frac{542553813}{159851548}a^{4}+\frac{81993399}{39962887}a^{3}-\frac{61045076}{39962887}a^{2}-\frac{135701417}{79925774}a+\frac{210572777}{79925774}$, $\frac{222554367}{159851548}a^{19}-\frac{509168275}{79925774}a^{18}+\frac{511580867}{39962887}a^{17}-\frac{500494676}{39962887}a^{16}+\frac{829906205}{159851548}a^{15}+\frac{146533366}{39962887}a^{14}-\frac{1033119016}{39962887}a^{13}+\frac{3913816719}{79925774}a^{12}-\frac{1352860415}{39962887}a^{11}-\frac{691539709}{39962887}a^{10}+\frac{4547026833}{79925774}a^{9}-\frac{8055926171}{79925774}a^{8}+\frac{26395597275}{159851548}a^{7}-\frac{16749898995}{79925774}a^{6}+\frac{6708584378}{39962887}a^{5}-\frac{3601532390}{39962887}a^{4}+\frac{4303612129}{79925774}a^{3}-\frac{2389800720}{39962887}a^{2}+\frac{1809281912}{39962887}a-\frac{673739279}{39962887}$, $\frac{185441261}{159851548}a^{19}-\frac{191087995}{39962887}a^{18}+\frac{379852603}{39962887}a^{17}-\frac{362963462}{39962887}a^{16}+\frac{691469655}{159851548}a^{15}+\frac{286580027}{79925774}a^{14}-\frac{778135360}{39962887}a^{13}+\frac{2824494883}{79925774}a^{12}-\frac{996824878}{39962887}a^{11}-\frac{482705279}{39962887}a^{10}+\frac{3524894015}{79925774}a^{9}-\frac{6045157777}{79925774}a^{8}+\frac{19619586033}{159851548}a^{7}-\frac{6203913785}{39962887}a^{6}+\frac{4988230097}{39962887}a^{5}-\frac{2752887800}{39962887}a^{4}+\frac{3204682633}{79925774}a^{3}-\frac{1778586105}{39962887}a^{2}+\frac{1392411566}{39962887}a-\frac{525169613}{39962887}$, $\frac{14437339}{79925774}a^{19}-\frac{44205651}{79925774}a^{18}+\frac{67991161}{79925774}a^{17}-\frac{20974243}{39962887}a^{16}+\frac{23149809}{79925774}a^{15}+\frac{40019785}{79925774}a^{14}-\frac{195438763}{79925774}a^{13}+\frac{107300367}{39962887}a^{12}-\frac{39412672}{39962887}a^{11}-\frac{69808256}{39962887}a^{10}+\frac{149417603}{39962887}a^{9}-\frac{313815303}{39962887}a^{8}+\frac{932969557}{79925774}a^{7}-\frac{1010484297}{79925774}a^{6}+\frac{675861359}{79925774}a^{5}-\frac{227656290}{39962887}a^{4}+\frac{142872432}{39962887}a^{3}-\frac{141489406}{39962887}a^{2}+\frac{85606645}{39962887}a-\frac{54674945}{39962887}$, $\frac{232642523}{159851548}a^{19}-\frac{760194913}{159851548}a^{18}+\frac{1222649677}{159851548}a^{17}-\frac{386336437}{79925774}a^{16}+\frac{327239455}{159851548}a^{15}+\frac{771020779}{159851548}a^{14}-\frac{3258143739}{159851548}a^{13}+\frac{2118975459}{79925774}a^{12}-\frac{304430169}{39962887}a^{11}-\frac{1298913857}{79925774}a^{10}+\frac{2947513139}{79925774}a^{9}-\frac{2588042393}{39962887}a^{8}+\frac{16239413287}{159851548}a^{7}-\frac{17405042763}{159851548}a^{6}+\frac{11503890233}{159851548}a^{5}-\frac{2967878963}{79925774}a^{4}+\frac{1249839015}{39962887}a^{3}-\frac{2335133289}{79925774}a^{2}+\frac{1398106427}{79925774}a-\frac{155071930}{39962887}$ 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:  \( 34612.1677943 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 34612.1677943 \cdot 1}{4\cdot\sqrt{1744047395068446282612736}}\cr\approx \mathstrut & 0.628330803542 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*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^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*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^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*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^20 - 4*x^19 + 8*x^18 - 8*x^17 + 5*x^16 + 2*x^15 - 16*x^14 + 30*x^13 - 22*x^12 - 4*x^11 + 34*x^10 - 66*x^9 + 109*x^8 - 136*x^7 + 120*x^6 - 76*x^5 + 48*x^4 - 40*x^3 + 32*x^2 - 16*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

$C_2\times A_5$ (as 20T31):

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 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.2.82538991616.1

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 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 10.0.1320623865856.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.3.0.1}{3} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.5.0.1}{5} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.3.0.1}{3} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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 2.8.14.2$x^{8} + 2 x^{7} + 2$$8$$1$$14$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.18.51$x^{12} + 2 x^{11} + 16 x^{10} + 44 x^{9} + 18 x^{8} - 8 x^{7} + 24 x^{6} + 40 x^{5} + 20 x^{4} + 8 x^{3} + 8$$4$$3$$18$$A_4 \times C_2$$[2, 2, 2]^{3}$
\(67\) Copy content Toggle raw display 67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$