Properties

Label 22.10.565...896.1
Degree $22$
Signature $[10, 6]$
Discriminant $5.650\times 10^{37}$
Root discriminant \(52.00\)
Ramified primes $2,1297$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.D_{11}$ (as 22T30)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^20 - 97*x^18 - 162*x^16 + 392*x^14 + 639*x^12 - 299*x^10 - 637*x^8 - 27*x^6 + 178*x^4 + 43*x^2 - 1)
 
gp: K = bnfinit(y^22 - 5*y^20 - 97*y^18 - 162*y^16 + 392*y^14 + 639*y^12 - 299*y^10 - 637*y^8 - 27*y^6 + 178*y^4 + 43*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 5*x^20 - 97*x^18 - 162*x^16 + 392*x^14 + 639*x^12 - 299*x^10 - 637*x^8 - 27*x^6 + 178*x^4 + 43*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 5*x^20 - 97*x^18 - 162*x^16 + 392*x^14 + 639*x^12 - 299*x^10 - 637*x^8 - 27*x^6 + 178*x^4 + 43*x^2 - 1)
 

\( x^{22} - 5 x^{20} - 97 x^{18} - 162 x^{16} + 392 x^{14} + 639 x^{12} - 299 x^{10} - 637 x^{8} - 27 x^{6} + 178 x^{4} + 43 x^{2} - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[10, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(56501459388151144478039723653407440896\) \(\medspace = 2^{22}\cdot 1297^{10}\) 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:  \(52.00\)
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:  not computed
Ramified primes:   \(2\), \(1297\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17}a^{18}-\frac{7}{17}a^{16}+\frac{5}{17}a^{12}+\frac{8}{17}a^{10}+\frac{1}{17}a^{8}+\frac{6}{17}a^{6}-\frac{5}{17}a^{4}+\frac{5}{17}a^{2}+\frac{8}{17}$, $\frac{1}{17}a^{19}-\frac{7}{17}a^{17}+\frac{5}{17}a^{13}+\frac{8}{17}a^{11}+\frac{1}{17}a^{9}+\frac{6}{17}a^{7}-\frac{5}{17}a^{5}+\frac{5}{17}a^{3}+\frac{8}{17}a$, $\frac{1}{47211049775}a^{20}+\frac{1315605556}{47211049775}a^{18}-\frac{4226733506}{47211049775}a^{16}-\frac{9396970128}{47211049775}a^{14}+\frac{15573620034}{47211049775}a^{12}-\frac{21814315437}{47211049775}a^{10}+\frac{12444220369}{47211049775}a^{8}-\frac{12396622753}{47211049775}a^{6}-\frac{3304348862}{9442209955}a^{4}-\frac{20220030807}{47211049775}a^{2}-\frac{372193984}{47211049775}$, $\frac{1}{47211049775}a^{21}+\frac{1315605556}{47211049775}a^{19}-\frac{4226733506}{47211049775}a^{17}-\frac{9396970128}{47211049775}a^{15}+\frac{15573620034}{47211049775}a^{13}-\frac{21814315437}{47211049775}a^{11}+\frac{12444220369}{47211049775}a^{9}-\frac{12396622753}{47211049775}a^{7}-\frac{3304348862}{9442209955}a^{5}-\frac{20220030807}{47211049775}a^{3}-\frac{372193984}{47211049775}a$ 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:  $15$
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:   $\frac{774460659}{2777120575}a^{20}-\frac{71517526107}{47211049775}a^{18}-\frac{1245452075993}{47211049775}a^{16}-\frac{94068610302}{2777120575}a^{14}+\frac{5784747450752}{47211049775}a^{12}+\frac{5863905461064}{47211049775}a^{10}-\frac{6092427956418}{47211049775}a^{8}-\frac{5572363655434}{47211049775}a^{6}+\frac{316197704769}{9442209955}a^{4}+\frac{1433176984329}{47211049775}a^{2}+\frac{66496391798}{47211049775}$, $\frac{1943506488}{2777120575}a^{21}-\frac{182386365924}{47211049775}a^{19}-\frac{3105867888876}{47211049775}a^{17}-\frac{221111286014}{2777120575}a^{15}+\frac{14521891812564}{47211049775}a^{13}+\frac{13055069475473}{47211049775}a^{11}-\frac{15030396500051}{47211049775}a^{9}-\frac{11690505477913}{47211049775}a^{7}+\frac{763403608493}{9442209955}a^{5}+\frac{2972012989803}{47211049775}a^{3}+\frac{251763951436}{47211049775}a$, $a$, $\frac{37171557421}{47211049775}a^{21}-\frac{180195676449}{47211049775}a^{19}-\frac{3647697995376}{47211049775}a^{17}-\frac{6487568931413}{47211049775}a^{15}+\frac{14900475826064}{47211049775}a^{13}+\frac{26849876452598}{47211049775}a^{11}-\frac{14043788707826}{47211049775}a^{9}-\frac{27296334019563}{47211049775}a^{7}+\frac{540484949968}{9442209955}a^{5}+\frac{7676389678778}{47211049775}a^{3}+\frac{450450605436}{47211049775}a$, $\frac{230966623}{47211049775}a^{20}+\frac{17988889}{2777120575}a^{18}-\frac{30736961063}{47211049775}a^{16}-\frac{171061837294}{47211049775}a^{14}-\frac{63532521593}{47211049775}a^{12}+\frac{625289087249}{47211049775}a^{10}+\frac{253078251987}{47211049775}a^{8}+\frac{22967031056}{47211049775}a^{6}-\frac{35909802706}{9442209955}a^{4}-\frac{371945105986}{47211049775}a^{2}-\frac{83728814057}{47211049775}$, $\frac{7132192536}{47211049775}a^{20}-\frac{43030735434}{47211049775}a^{18}-\frac{648813296566}{47211049775}a^{16}-\frac{476464586658}{47211049775}a^{14}+\frac{3421874674174}{47211049775}a^{12}+\frac{1142605175893}{47211049775}a^{10}-\frac{3980745096166}{47211049775}a^{8}-\frac{781171245108}{47211049775}a^{6}+\frac{251382231478}{9442209955}a^{4}+\frac{234242896073}{47211049775}a^{2}+\frac{18847362626}{47211049775}$, $\frac{18965113729}{47211049775}a^{21}-\frac{104859003776}{47211049775}a^{19}-\frac{1781688243899}{47211049775}a^{17}-\frac{2144509562437}{47211049775}a^{15}+\frac{8345351561486}{47211049775}a^{13}+\frac{7537040101577}{47211049775}a^{11}-\frac{8464623494949}{47211049775}a^{9}-\frac{7145453826262}{47211049775}a^{7}+\frac{367384899332}{9442209955}a^{5}+\frac{1895600887972}{47211049775}a^{3}+\frac{210181612939}{47211049775}a$, $\frac{40790389434}{47211049775}a^{20}-\frac{227342352796}{47211049775}a^{18}-\frac{3825429258554}{47211049775}a^{16}-\frac{4419668599502}{47211049775}a^{14}+\frac{18444651090081}{47211049775}a^{12}+\frac{15426218669517}{47211049775}a^{10}-\frac{20637719799754}{47211049775}a^{8}-\frac{14001314306702}{47211049775}a^{6}+\frac{1310343548007}{9442209955}a^{4}+\frac{3514213486037}{47211049775}a^{2}-\frac{184155382281}{47211049775}$, $\frac{4670288748}{9442209955}a^{21}-\frac{24262363617}{9442209955}a^{19}-\frac{448193905378}{9442209955}a^{17}-\frac{669365010364}{9442209955}a^{15}+\frac{1950807268047}{9442209955}a^{13}+\frac{2565294168789}{9442209955}a^{11}-\frac{1878496984063}{9442209955}a^{9}-\frac{2439249870914}{9442209955}a^{7}+\frac{70022603333}{1888441991}a^{5}+\frac{632083310054}{9442209955}a^{3}+\frac{66308785108}{9442209955}a$, $\frac{4186460892}{47211049775}a^{21}-\frac{1949516169}{2777120575}a^{19}-\frac{334912334302}{47211049775}a^{17}+\frac{445359947474}{47211049775}a^{15}+\frac{2695281377353}{47211049775}a^{13}-\frac{2817513524654}{47211049775}a^{11}-\frac{4177554963652}{47211049775}a^{9}+\frac{2875125875674}{47211049775}a^{7}+\frac{395921186801}{9442209955}a^{5}-\frac{565837006044}{47211049775}a^{3}-\frac{187769442728}{47211049775}a$, $\frac{13103133954}{47211049775}a^{21}+\frac{18070005021}{47211049775}a^{20}-\frac{79491038801}{47211049775}a^{19}-\frac{107874431424}{47211049775}a^{18}-\frac{1191542665999}{47211049775}a^{17}-\frac{1651859116301}{47211049775}a^{16}-\frac{826059442612}{47211049775}a^{15}-\frac{1306912440288}{47211049775}a^{14}+\frac{6534696899511}{47211049775}a^{13}+\frac{8706215975639}{47211049775}a^{12}+\frac{2359588873502}{47211049775}a^{11}+\frac{3710195011048}{47211049775}a^{10}-\frac{476250887697}{2777120575}a^{9}-\frac{10244519847876}{47211049775}a^{8}-\frac{2925589222487}{47211049775}a^{7}-\frac{3616750540913}{47211049775}a^{6}+\frac{568607477232}{9442209955}a^{5}+\frac{666294301883}{9442209955}a^{4}+\frac{1260085633072}{47211049775}a^{3}+\frac{1298023511828}{47211049775}a^{2}+\frac{52722499289}{47211049775}a+\frac{54715259661}{47211049775}$, $\frac{17447116274}{47211049775}a^{21}+\frac{5276420893}{47211049775}a^{20}-\frac{93454813106}{47211049775}a^{19}-\frac{33208642317}{47211049775}a^{18}-\frac{1660220872194}{47211049775}a^{17}-\frac{476208533958}{47211049775}a^{16}-\frac{2232802790872}{47211049775}a^{15}-\frac{204092298254}{47211049775}a^{14}+\frac{7767287184866}{47211049775}a^{13}+\frac{179565263811}{2777120575}a^{12}+\frac{8956765560587}{47211049775}a^{11}+\frac{852772348934}{47211049775}a^{10}-\frac{8133490426994}{47211049775}a^{9}-\frac{4725663739208}{47211049775}a^{8}-\frac{10134523588997}{47211049775}a^{7}-\frac{1820155314179}{47211049775}a^{6}+\frac{227857164772}{9442209955}a^{5}+\frac{353419909149}{9442209955}a^{4}+\frac{3304932603257}{47211049775}a^{3}+\frac{54903149297}{2777120575}a^{2}+\frac{770837452234}{47211049775}a+\frac{101264702663}{47211049775}$, $\frac{1227880353723}{47211049775}a^{21}-\frac{170066131033}{47211049775}a^{20}-\frac{6097154537337}{47211049775}a^{19}+\frac{837470078352}{47211049775}a^{18}-\frac{119328077595963}{47211049775}a^{17}+\frac{16567978479623}{47211049775}a^{16}-\frac{202941811003144}{47211049775}a^{15}+\frac{28756728901749}{47211049775}a^{14}+\frac{475630722250082}{47211049775}a^{13}-\frac{65256765233847}{47211049775}a^{12}+\frac{802221510576349}{47211049775}a^{11}-\frac{114414054907529}{47211049775}a^{10}-\frac{345905995813438}{47211049775}a^{9}+\frac{45916355327198}{47211049775}a^{8}-\frac{797850097839219}{47211049775}a^{7}+\frac{114375255012199}{47211049775}a^{6}-\frac{632136112763}{555424115}a^{5}+\frac{1846656096641}{9442209955}a^{4}+\frac{219929920230189}{47211049775}a^{3}-\frac{31678457516919}{47211049775}a^{2}+\frac{58454540883093}{47211049775}a-\frac{8556646290028}{47211049775}$, $\frac{15245364518}{47211049775}a^{21}+\frac{1193979632}{2777120575}a^{20}-\frac{72767922992}{47211049775}a^{19}-\frac{119440050061}{47211049775}a^{18}-\frac{1498102048883}{47211049775}a^{17}-\frac{1865031531789}{47211049775}a^{16}-\frac{2794184224154}{47211049775}a^{15}-\frac{95842272621}{2777120575}a^{14}+\frac{5603755610087}{47211049775}a^{13}+\frac{9567493649521}{47211049775}a^{12}+\frac{11329481774509}{47211049775}a^{11}+\frac{4765830514622}{47211049775}a^{10}-\frac{3152497054533}{47211049775}a^{9}-\frac{10853573246814}{47211049775}a^{8}-\frac{11343058446529}{47211049775}a^{7}-\frac{3852478315657}{47211049775}a^{6}-\frac{350370071576}{9442209955}a^{5}+\frac{669993744592}{9442209955}a^{4}+\frac{2999285394749}{47211049775}a^{3}+\frac{817808342742}{47211049775}a^{2}+\frac{901915469138}{47211049775}a-\frac{163761939771}{47211049775}$, $\frac{67385927896}{47211049775}a^{21}-\frac{6563583061}{47211049775}a^{20}-\frac{357926419949}{47211049775}a^{19}+\frac{3979543602}{2777120575}a^{18}-\frac{6392917272776}{47211049775}a^{17}+\frac{465048370016}{47211049775}a^{16}-\frac{9075487227388}{47211049775}a^{15}-\frac{2304976593942}{47211049775}a^{14}+\frac{26118701316839}{47211049775}a^{13}-\frac{8573493325449}{47211049775}a^{12}+\frac{28756676018898}{47211049775}a^{11}+\frac{6757191347007}{47211049775}a^{10}-\frac{20387186291926}{47211049775}a^{9}+\frac{20648123687441}{47211049775}a^{8}-\frac{17742659956738}{47211049775}a^{7}+\frac{1694853157508}{47211049775}a^{6}+\frac{1072549913068}{9442209955}a^{5}-\frac{1738130436273}{9442209955}a^{4}+\frac{2917530801303}{47211049775}a^{3}-\frac{2129578526073}{47211049775}a^{2}-\frac{118182300889}{47211049775}a+\frac{150308277474}{47211049775}$ 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:  \( 44489140059.3 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{6}\cdot 44489140059.3 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.186454706213 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^20 - 97*x^18 - 162*x^16 + 392*x^14 + 639*x^12 - 299*x^10 - 637*x^8 - 27*x^6 + 178*x^4 + 43*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^22 - 5*x^20 - 97*x^18 - 162*x^16 + 392*x^14 + 639*x^12 - 299*x^10 - 637*x^8 - 27*x^6 + 178*x^4 + 43*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^22 - 5*x^20 - 97*x^18 - 162*x^16 + 392*x^14 + 639*x^12 - 299*x^10 - 637*x^8 - 27*x^6 + 178*x^4 + 43*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^22 - 5*x^20 - 97*x^18 - 162*x^16 + 392*x^14 + 639*x^12 - 299*x^10 - 637*x^8 - 27*x^6 + 178*x^4 + 43*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

$C_2^{10}.D_{11}$ (as 22T30):

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 22528
The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ are not computed
Character table for $C_2^{10}.D_{11}$ is not computed

Intermediate fields

11.11.3670285774226257.1

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 22 siblings: data not computed
Degree 44 siblings: data not computed
Minimal sibling: 22.10.73282392826432034388017521578469450842112.6

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{5}$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{5}$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.11.0.1}{11} }^{2}$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.11.0.1}{11} }^{2}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{9}$

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 Deg $22$$2$$11$$22$
\(1297\) Copy content Toggle raw display $\Q_{1297}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1297}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$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}$
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$