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Properties

Label 20.14.956...016.1
Degree $20$
Signature $[14, 3]$
Discriminant $-9.561\times 10^{31}$
Root discriminant \(39.72\)
Ramified primes $2,11,113$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.C_2^4:A_5$ (as 20T968)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 17*x^18 + 63*x^16 + 31*x^14 - 545*x^12 + 878*x^10 - 104*x^8 - 712*x^6 + 464*x^4 - 48*x^2 - 16)
 
gp: K = bnfinit(y^20 - 17*y^18 + 63*y^16 + 31*y^14 - 545*y^12 + 878*y^10 - 104*y^8 - 712*y^6 + 464*y^4 - 48*y^2 - 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 17*x^18 + 63*x^16 + 31*x^14 - 545*x^12 + 878*x^10 - 104*x^8 - 712*x^6 + 464*x^4 - 48*x^2 - 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 17*x^18 + 63*x^16 + 31*x^14 - 545*x^12 + 878*x^10 - 104*x^8 - 712*x^6 + 464*x^4 - 48*x^2 - 16)
 

\( x^{20} - 17x^{18} + 63x^{16} + 31x^{14} - 545x^{12} + 878x^{10} - 104x^{8} - 712x^{6} + 464x^{4} - 48x^{2} - 16 \) 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:  $[14, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-95606831403792437215606148694016\) \(\medspace = -\,2^{24}\cdot 11^{8}\cdot 113^{8}\) 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:  \(39.72\)
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:  $2^{247/96}11^{2/3}113^{2/3}\approx 687.8691707033878$
Ramified primes:   \(2\), \(11\), \(113\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{6}$, $\frac{1}{8}a^{17}-\frac{1}{8}a^{15}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{7}$, $\frac{1}{574088}a^{18}-\frac{52411}{574088}a^{16}+\frac{15571}{574088}a^{14}+\frac{95627}{574088}a^{12}-\frac{72085}{574088}a^{10}+\frac{20469}{287044}a^{8}+\frac{15307}{287044}a^{6}-\frac{34475}{71761}a^{4}-\frac{12923}{71761}a^{2}+\frac{22621}{71761}$, $\frac{1}{574088}a^{19}+\frac{9675}{287044}a^{17}-\frac{28095}{287044}a^{15}+\frac{11933}{287044}a^{13}+\frac{71599}{287044}a^{11}-\frac{30823}{574088}a^{9}-\frac{28227}{143522}a^{7}-\frac{1}{2}a^{6}-\frac{34475}{71761}a^{5}-\frac{12923}{71761}a^{3}+\frac{22621}{71761}a$ 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:  $16$
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:   $\frac{903135}{574088}a^{19}-\frac{14861563}{574088}a^{17}+\frac{48796439}{574088}a^{15}+\frac{54696071}{574088}a^{13}-\frac{462617549}{574088}a^{11}+\frac{135049635}{143522}a^{9}+\frac{101144371}{287044}a^{7}-\frac{133523633}{143522}a^{5}+\frac{32068037}{143522}a^{3}+\frac{3550512}{71761}a$, $\frac{243}{287044}a^{19}-\frac{140113}{574088}a^{17}+\frac{1898387}{574088}a^{15}-\frac{2825085}{574088}a^{13}-\frac{14868469}{574088}a^{11}+\frac{34893917}{574088}a^{9}+\frac{131551}{143522}a^{7}-\frac{3981392}{71761}a^{5}+\frac{1110805}{71761}a^{3}+\frac{157895}{71761}a$, $\frac{103023}{287044}a^{19}-\frac{1819555}{287044}a^{17}+\frac{7345361}{287044}a^{15}+\frac{3300781}{287044}a^{13}-\frac{66174229}{287044}a^{11}+\frac{24008575}{71761}a^{9}+\frac{9569299}{143522}a^{7}-\frac{45053633}{143522}a^{5}+\frac{13051157}{143522}a^{3}+\frac{1094270}{71761}a$, $\frac{274453}{287044}a^{19}-\frac{9128157}{574088}a^{17}+\frac{31065695}{574088}a^{15}+\frac{30431427}{574088}a^{13}-\frac{290725597}{574088}a^{11}+\frac{359325137}{574088}a^{9}+\frac{13823797}{71761}a^{7}-\frac{87219871}{143522}a^{5}+\frac{11831416}{71761}a^{3}+\frac{2006104}{71761}a$, $\frac{53817}{71761}a^{18}-\frac{897814}{71761}a^{16}+\frac{6162305}{143522}a^{14}+\frac{5992451}{143522}a^{12}-\frac{57908697}{143522}a^{10}+\frac{70736355}{143522}a^{8}+\frac{24743223}{143522}a^{6}-\frac{34702728}{71761}a^{4}+\frac{8066117}{71761}a^{2}+\frac{1833005}{71761}$, $\frac{288487}{574088}a^{18}-\frac{4585683}{574088}a^{16}+\frac{13283545}{574088}a^{14}+\frac{21162853}{574088}a^{12}-\frac{130139659}{574088}a^{10}+\frac{64269091}{287044}a^{8}+\frac{8374500}{71761}a^{6}-\frac{32685359}{143522}a^{4}+\frac{3598021}{71761}a^{2}+\frac{780219}{71761}$, $\frac{243}{574088}a^{19}-\frac{7329}{71761}a^{18}-\frac{4272}{71761}a^{17}+\frac{436671}{287044}a^{16}+\frac{205493}{287044}a^{15}-\frac{235152}{71761}a^{14}-\frac{114243}{287044}a^{13}-\frac{678206}{71761}a^{12}-\frac{1976913}{287044}a^{11}+\frac{2518118}{71761}a^{10}+\frac{4852903}{574088}a^{9}-\frac{3089445}{143522}a^{8}+\frac{3289035}{287044}a^{7}-\frac{6997775}{287044}a^{6}-\frac{1756801}{143522}a^{5}+\frac{4182603}{143522}a^{4}-\frac{413371}{71761}a^{3}-\frac{238346}{71761}a^{2}+\frac{258350}{71761}a-\frac{242953}{71761}$, $\frac{2801}{574088}a^{19}+\frac{1279}{287044}a^{18}-\frac{123727}{574088}a^{17}-\frac{8895}{287044}a^{16}+\frac{1275381}{574088}a^{15}-\frac{53005}{143522}a^{14}-\frac{1826611}{574088}a^{13}+\frac{192497}{143522}a^{12}-\frac{9734127}{574088}a^{11}+\frac{295129}{143522}a^{10}+\frac{3049000}{71761}a^{9}-\frac{2824507}{287044}a^{8}-\frac{458049}{71761}a^{7}+\frac{1193737}{287044}a^{6}-\frac{5043369}{143522}a^{5}+\frac{581307}{71761}a^{4}+\frac{1190158}{71761}a^{3}-\frac{334018}{71761}a^{2}-\frac{3542}{71761}a+\frac{25152}{71761}$, $\frac{1020069}{574088}a^{19}+\frac{171605}{143522}a^{18}-\frac{16822301}{574088}a^{17}-\frac{1419341}{71761}a^{16}+\frac{55601195}{574088}a^{15}+\frac{18954383}{287044}a^{14}+\frac{61403159}{574088}a^{13}+\frac{20270639}{287044}a^{12}-\frac{526912213}{574088}a^{11}-\frac{179157707}{287044}a^{10}+\frac{307922969}{287044}a^{9}+\frac{211867979}{287044}a^{8}+\frac{29262357}{71761}a^{7}+\frac{78363615}{287044}a^{6}-\frac{151571311}{143522}a^{5}-\frac{104108359}{143522}a^{4}+\frac{17653697}{71761}a^{3}+\frac{12257964}{71761}a^{2}+\frac{3604066}{71761}a+\frac{2588558}{71761}$, $\frac{96887}{574088}a^{19}-\frac{35697}{574088}a^{18}-\frac{1714939}{574088}a^{17}+\frac{823807}{574088}a^{16}+\frac{6956791}{574088}a^{15}-\frac{5287595}{574088}a^{14}+\frac{3107923}{574088}a^{13}+\frac{3087713}{574088}a^{12}-\frac{63037989}{574088}a^{11}+\frac{44073561}{574088}a^{10}+\frac{11354430}{71761}a^{9}-\frac{42063337}{287044}a^{8}+\frac{10800633}{287044}a^{7}-\frac{2609121}{287044}a^{6}-\frac{10971252}{71761}a^{5}+\frac{18922515}{143522}a^{4}+\frac{2670384}{71761}a^{3}-\frac{2766056}{71761}a^{2}+\frac{745736}{71761}a-\frac{549392}{71761}$, $\frac{165615}{574088}a^{19}+\frac{593339}{574088}a^{18}-\frac{658971}{143522}a^{17}-\frac{9763997}{574088}a^{16}+\frac{958287}{71761}a^{15}+\frac{32088791}{574088}a^{14}+\frac{1521385}{71761}a^{13}+\frac{35552139}{574088}a^{12}-\frac{9398445}{71761}a^{11}-\frac{303399625}{574088}a^{10}+\frac{73666137}{574088}a^{9}+\frac{89490501}{143522}a^{8}+\frac{10030519}{143522}a^{7}+\frac{15811458}{71761}a^{6}-\frac{18412419}{143522}a^{5}-\frac{87828697}{143522}a^{4}+\frac{3431127}{143522}a^{3}+\frac{10994147}{71761}a^{2}+\frac{309193}{71761}a+\frac{2183953}{71761}$, $\frac{165615}{574088}a^{19}-\frac{593339}{574088}a^{18}-\frac{658971}{143522}a^{17}+\frac{9763997}{574088}a^{16}+\frac{958287}{71761}a^{15}-\frac{32088791}{574088}a^{14}+\frac{1521385}{71761}a^{13}-\frac{35552139}{574088}a^{12}-\frac{9398445}{71761}a^{11}+\frac{303399625}{574088}a^{10}+\frac{73666137}{574088}a^{9}-\frac{89490501}{143522}a^{8}+\frac{10030519}{143522}a^{7}-\frac{15811458}{71761}a^{6}-\frac{18412419}{143522}a^{5}+\frac{87828697}{143522}a^{4}+\frac{3431127}{143522}a^{3}-\frac{10994147}{71761}a^{2}+\frac{309193}{71761}a-\frac{2183953}{71761}$, $\frac{460813}{574088}a^{18}-\frac{7681693}{574088}a^{16}+\frac{26344881}{574088}a^{14}+\frac{25147353}{574088}a^{12}-\frac{246539567}{574088}a^{10}+\frac{76734357}{143522}a^{8}+\frac{48068727}{287044}a^{6}-\frac{37680759}{71761}a^{4}+\frac{9831443}{71761}a^{2}+\frac{1985560}{71761}$, $\frac{354229}{574088}a^{19}+\frac{157015}{143522}a^{18}-\frac{5805167}{574088}a^{17}-\frac{4905445}{287044}a^{16}+\frac{18807159}{574088}a^{15}+\frac{13242375}{287044}a^{14}+\frac{21896531}{574088}a^{13}+\frac{24822207}{287044}a^{12}-\frac{178852769}{574088}a^{11}-\frac{131957101}{287044}a^{10}+\frac{103030757}{287044}a^{9}+\frac{118254401}{287044}a^{8}+\frac{39534215}{287044}a^{7}+\frac{17805921}{71761}a^{6}-\frac{25376472}{71761}a^{5}-\frac{30922722}{71761}a^{4}+\frac{12280299}{143522}a^{3}+\frac{6638776}{71761}a^{2}+\frac{1185603}{71761}a+\frac{1394178}{71761}$, $\frac{180933}{143522}a^{19}-\frac{210657}{287044}a^{18}-\frac{11785515}{574088}a^{17}+\frac{3615183}{287044}a^{16}+\frac{37414535}{574088}a^{15}-\frac{6753833}{143522}a^{14}+\frac{45570491}{574088}a^{13}-\frac{2323308}{71761}a^{12}-\frac{355693497}{574088}a^{11}+\frac{62041463}{143522}a^{10}+\frac{408137831}{574088}a^{9}-\frac{166770655}{287044}a^{8}+\frac{72827003}{287044}a^{7}-\frac{43881821}{287044}a^{6}-\frac{50167788}{71761}a^{5}+\frac{79756361}{143522}a^{4}+\frac{13457084}{71761}a^{3}-\frac{10558637}{71761}a^{2}+\frac{2139862}{71761}a-\frac{2046653}{71761}$, $\frac{476659}{574088}a^{19}-\frac{7768107}{574088}a^{17}+\frac{24646365}{574088}a^{15}+\frac{31031921}{574088}a^{13}-\frac{236747383}{574088}a^{11}+\frac{130712531}{287044}a^{9}+\frac{29381829}{143522}a^{7}-\frac{66483629}{143522}a^{5}+\frac{6558473}{71761}a^{3}+\frac{2278775}{71761}a+1$ 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:  \( 2796991987.83 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{3}\cdot 2796991987.83 \cdot 1}{2\cdot\sqrt{95606831403792437215606148694016}}\cr\approx \mathstrut & 0.581267857686 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 17*x^18 + 63*x^16 + 31*x^14 - 545*x^12 + 878*x^10 - 104*x^8 - 712*x^6 + 464*x^4 - 48*x^2 - 16)
 
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 - 17*x^18 + 63*x^16 + 31*x^14 - 545*x^12 + 878*x^10 - 104*x^8 - 712*x^6 + 464*x^4 - 48*x^2 - 16, 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 - 17*x^18 + 63*x^16 + 31*x^14 - 545*x^12 + 878*x^10 - 104*x^8 - 712*x^6 + 464*x^4 - 48*x^2 - 16);
 
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 - 17*x^18 + 63*x^16 + 31*x^14 - 545*x^12 + 878*x^10 - 104*x^8 - 712*x^6 + 464*x^4 - 48*x^2 - 16);
 
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

$C_2^{10}.C_2^4:A_5$ (as 20T968):

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 983040
The 188 conjugacy class representatives for $C_2^{10}.C_2^4:A_5$
Character table for $C_2^{10}.C_2^4:A_5$

Intermediate fields

5.5.6180196.1, 10.10.152779290393664.1

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 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.8.382427325615169748862424594776064.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ R ${\href{/padicField/13.5.0.1}{5} }^{4}$ ${\href{/padicField/17.5.0.1}{5} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$

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 2.4.0.1$x^{4} + x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} + x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.24.388$x^{12} + 4 x^{11} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 6 x^{2} + 4 x + 14$$12$$1$$24$12T149$[4/3, 4/3, 2, 7/3, 7/3, 3]_{3}^{2}$
\(11\) Copy content Toggle raw display 11.4.0.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.12.8.1$x^{12} + 24 x^{10} + 74 x^{9} + 198 x^{8} + 480 x^{7} - 346 x^{6} + 5208 x^{5} - 3276 x^{4} - 856 x^{3} + 53628 x^{2} - 74328 x + 152743$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
\(113\) Copy content Toggle raw display 113.4.0.1$x^{4} + 62 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.0.1$x^{4} + 62 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
113.6.4.1$x^{6} + 303 x^{5} + 30612 x^{4} + 1032345 x^{3} + 126075 x^{2} + 3458832 x + 116334092$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
113.6.4.1$x^{6} + 303 x^{5} + 30612 x^{4} + 1032345 x^{3} + 126075 x^{2} + 3458832 x + 116334092$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$