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Properties

Label 20.0.227...281.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.276\times 10^{31}$
Root discriminant \(36.97\)
Ramified prime $8311$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $C_2\times A_5$ (as 20T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024)
 
gp: K = bnfinit(y^20 - 3*y^19 + 10*y^18 - 26*y^17 + 59*y^16 - 129*y^15 + 241*y^14 - 438*y^13 + 729*y^12 - 1124*y^11 + 1669*y^10 - 2248*y^9 + 2916*y^8 - 3504*y^7 + 3856*y^6 - 4128*y^5 + 3776*y^4 - 3328*y^3 + 2560*y^2 - 1536*y + 1024, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024)
 

\( x^{20} - 3 x^{19} + 10 x^{18} - 26 x^{17} + 59 x^{16} - 129 x^{15} + 241 x^{14} - 438 x^{13} + \cdots + 1024 \) 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:   \(22762830184956456665916278633281\) \(\medspace = 8311^{8}\) 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:  \(36.97\)
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:  $8311^{1/2}\approx 91.16468614545876$
Ramified primes:   \(8311\) 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 a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{10}-\frac{1}{8}a^{9}-\frac{3}{8}a^{8}+\frac{3}{8}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{8}a^{11}+\frac{7}{16}a^{10}-\frac{3}{16}a^{9}+\frac{3}{16}a^{8}-\frac{1}{2}a^{7}+\frac{1}{16}a^{6}-\frac{1}{8}a^{5}-\frac{7}{16}a^{4}-\frac{3}{8}a^{3}$, $\frac{1}{32}a^{15}-\frac{1}{32}a^{14}-\frac{1}{16}a^{12}+\frac{7}{32}a^{11}-\frac{3}{32}a^{10}-\frac{13}{32}a^{9}-\frac{1}{4}a^{8}+\frac{1}{32}a^{7}+\frac{7}{16}a^{6}-\frac{7}{32}a^{5}+\frac{5}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{16}-\frac{1}{64}a^{15}-\frac{1}{32}a^{13}+\frac{7}{64}a^{12}-\frac{3}{64}a^{11}+\frac{19}{64}a^{10}-\frac{1}{8}a^{9}-\frac{31}{64}a^{8}-\frac{9}{32}a^{7}-\frac{7}{64}a^{6}+\frac{5}{32}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{17}-\frac{1}{128}a^{16}-\frac{1}{64}a^{14}+\frac{7}{128}a^{13}-\frac{3}{128}a^{12}+\frac{19}{128}a^{11}-\frac{1}{16}a^{10}+\frac{33}{128}a^{9}-\frac{9}{64}a^{8}-\frac{7}{128}a^{7}-\frac{27}{64}a^{6}+\frac{3}{8}a^{5}-\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{256}a^{18}-\frac{1}{256}a^{17}-\frac{1}{128}a^{15}+\frac{7}{256}a^{14}-\frac{3}{256}a^{13}+\frac{19}{256}a^{12}-\frac{1}{32}a^{11}+\frac{33}{256}a^{10}-\frac{9}{128}a^{9}-\frac{7}{256}a^{8}-\frac{27}{128}a^{7}-\frac{5}{16}a^{6}-\frac{1}{16}a^{5}+\frac{3}{16}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{49664}a^{19}-\frac{17}{49664}a^{18}-\frac{35}{12416}a^{17}-\frac{3}{24832}a^{16}-\frac{633}{49664}a^{15}+\frac{197}{49664}a^{14}-\frac{577}{49664}a^{13}-\frac{321}{12416}a^{12}+\frac{9005}{49664}a^{11}+\frac{1199}{24832}a^{10}-\frac{10563}{49664}a^{9}+\frac{6857}{24832}a^{8}-\frac{5851}{12416}a^{7}-\frac{609}{6208}a^{6}-\frac{637}{3104}a^{5}+\frac{31}{776}a^{4}+\frac{55}{388}a^{3}+\frac{87}{194}a^{2}-\frac{22}{97}a+\frac{14}{97}$ 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

$C_{4}$, which has order $4$ (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:  $9$
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:   $\frac{1231}{49664}a^{19}-\frac{5019}{49664}a^{18}+\frac{1689}{6208}a^{17}-\frac{16109}{24832}a^{16}+\frac{67393}{49664}a^{15}-\frac{127257}{49664}a^{14}+\frac{216645}{49664}a^{13}-\frac{20921}{3104}a^{12}+\frac{484995}{49664}a^{11}-\frac{316397}{24832}a^{10}+\frac{753099}{49664}a^{9}-\frac{411447}{24832}a^{8}+\frac{192893}{12416}a^{7}-\frac{76111}{6208}a^{6}+\frac{20953}{3104}a^{5}-\frac{1569}{1552}a^{4}-\frac{3591}{776}a^{3}+\frac{3801}{388}a^{2}-\frac{1881}{194}a+\frac{356}{97}$, $\frac{1}{512}a^{19}-\frac{3}{512}a^{18}+\frac{5}{256}a^{17}-\frac{13}{256}a^{16}+\frac{59}{512}a^{15}-\frac{129}{512}a^{14}+\frac{241}{512}a^{13}-\frac{219}{256}a^{12}+\frac{729}{512}a^{11}-\frac{281}{128}a^{10}+\frac{1669}{512}a^{9}-\frac{281}{64}a^{8}+\frac{729}{128}a^{7}-\frac{219}{32}a^{6}+\frac{241}{32}a^{5}-\frac{129}{16}a^{4}+\frac{59}{8}a^{3}-\frac{13}{2}a^{2}+4a-2$, $\frac{3}{512}a^{19}-\frac{7}{512}a^{18}+\frac{3}{64}a^{17}-\frac{29}{256}a^{16}+\frac{125}{512}a^{15}-\frac{269}{512}a^{14}+\frac{465}{512}a^{13}-\frac{13}{8}a^{12}+\frac{1311}{512}a^{11}-\frac{957}{256}a^{10}+\frac{2759}{512}a^{9}-\frac{1703}{256}a^{8}+\frac{1063}{128}a^{7}-\frac{585}{64}a^{6}+\frac{285}{32}a^{5}-\frac{73}{8}a^{4}+6a^{3}-\frac{23}{4}a^{2}+2a-1$, $\frac{15}{6208}a^{19}+\frac{435}{24832}a^{18}-\frac{349}{24832}a^{17}+\frac{887}{12416}a^{16}-\frac{657}{12416}a^{15}+\frac{1441}{24832}a^{14}+\frac{1173}{24832}a^{13}-\frac{15057}{24832}a^{12}+\frac{16689}{12416}a^{11}-\frac{83585}{24832}a^{10}+\frac{9495}{1552}a^{9}-\frac{250077}{24832}a^{8}+\frac{49995}{3104}a^{7}-\frac{129049}{6208}a^{6}+\frac{43357}{1552}a^{5}-\frac{47947}{1552}a^{4}+\frac{5919}{194}a^{3}-\frac{3025}{97}a^{2}+\frac{3741}{194}a-\frac{1715}{97}$, $\frac{39}{24832}a^{19}+\frac{1}{1552}a^{18}+\frac{69}{24832}a^{17}-\frac{505}{12416}a^{16}+\frac{2667}{24832}a^{15}-\frac{1935}{6208}a^{14}+\frac{4729}{6208}a^{13}-\frac{36399}{24832}a^{12}+\frac{74939}{24832}a^{11}-\frac{120751}{24832}a^{10}+\frac{195069}{24832}a^{9}-\frac{291691}{24832}a^{8}+\frac{185665}{12416}a^{7}-\frac{15565}{776}a^{6}+\frac{33357}{1552}a^{5}-\frac{18049}{776}a^{4}+\frac{17019}{776}a^{3}-\frac{1554}{97}a^{2}+\frac{1388}{97}a-\frac{460}{97}$, $\frac{879}{49664}a^{19}-\frac{3109}{49664}a^{18}+\frac{4527}{24832}a^{17}-\frac{10979}{24832}a^{16}+\frac{43829}{49664}a^{15}-\frac{84663}{49664}a^{14}+\frac{139031}{49664}a^{13}-\frac{107545}{24832}a^{12}+\frac{303223}{49664}a^{11}-\frac{91657}{12416}a^{10}+\frac{420555}{49664}a^{9}-\frac{12015}{1552}a^{8}+\frac{68199}{12416}a^{7}-\frac{2223}{1552}a^{6}-\frac{18081}{3104}a^{5}+\frac{3973}{388}a^{4}-\frac{12629}{776}a^{3}+\frac{7543}{388}a^{2}-\frac{1684}{97}a+\frac{1345}{97}$, $\frac{993}{49664}a^{19}-\frac{2719}{49664}a^{18}+\frac{3725}{24832}a^{17}-\frac{9187}{24832}a^{16}+\frac{34523}{49664}a^{15}-\frac{68413}{49664}a^{14}+\frac{110113}{49664}a^{13}-\frac{83927}{24832}a^{12}+\frac{247661}{49664}a^{11}-\frac{37847}{6208}a^{10}+\frac{375329}{49664}a^{9}-\frac{100449}{12416}a^{8}+\frac{11175}{1552}a^{7}-\frac{5061}{776}a^{6}+\frac{10181}{3104}a^{5}-\frac{1581}{1552}a^{4}-\frac{671}{776}a^{3}+\frac{1771}{388}a^{2}-\frac{527}{194}a+\frac{128}{97}$, $\frac{61}{49664}a^{19}-\frac{455}{49664}a^{18}+\frac{95}{24832}a^{17}+\frac{981}{24832}a^{16}-\frac{4857}{49664}a^{15}+\frac{19195}{49664}a^{14}-\frac{44703}{49664}a^{13}+\frac{49011}{24832}a^{12}-\frac{191775}{49664}a^{11}+\frac{78813}{12416}a^{10}-\frac{524451}{49664}a^{9}+\frac{93875}{6208}a^{8}-\frac{63159}{3104}a^{7}+\frac{81675}{3104}a^{6}-\frac{43921}{1552}a^{5}+\frac{47723}{1552}a^{4}-\frac{21517}{776}a^{3}+\frac{2120}{97}a^{2}-\frac{1827}{97}a+\frac{272}{97}$, $\frac{1}{256}a^{19}-\frac{11}{256}a^{18}+\frac{17}{128}a^{17}-\frac{49}{128}a^{16}+\frac{251}{256}a^{15}-\frac{513}{256}a^{14}+\frac{1049}{256}a^{13}-\frac{923}{128}a^{12}+\frac{3033}{256}a^{11}-\frac{1203}{64}a^{10}+\frac{6621}{256}a^{9}-\frac{1135}{32}a^{8}+\frac{2803}{64}a^{7}-\frac{1565}{32}a^{6}+\frac{1743}{32}a^{5}-\frac{395}{8}a^{4}+\frac{355}{8}a^{3}-\frac{143}{4}a^{2}+\frac{33}{2}a-16$ 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:  \( 10106391.1313 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 10106391.1313 \cdot 4}{2\cdot\sqrt{22762830184956456665916278633281}}\cr\approx \mathstrut & 0.406267098100 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024)
 
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 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024, 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 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024);
 
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 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024);
 
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 20T36):

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

10.10.4771040786343841.1

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: 10.0.39652119975303662551.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{4}$ ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.3.0.1}{3} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.5.0.1}{5} }^{4}$ ${\href{/padicField/13.3.0.1}{3} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{10}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(8311\) Copy content Toggle raw display 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 $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$