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Properties

Label 22.14.719...104.3
Degree $22$
Signature $[14, 4]$
Discriminant $7.198\times 10^{33}$
Root discriminant \(34.59\)
Ramified primes $2,23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}:C_{11}$ (as 22T23)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^20 - 20*x^18 + 90*x^16 + 144*x^14 - 445*x^12 - 333*x^10 + 572*x^8 - 107*x^6 - 50*x^4 + 15*x^2 - 1)
 
Copy content gp:K = bnfinit(y^22 - 5*y^20 - 20*y^18 + 90*y^16 + 144*y^14 - 445*y^12 - 333*y^10 + 572*y^8 - 107*y^6 - 50*y^4 + 15*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 5*x^20 - 20*x^18 + 90*x^16 + 144*x^14 - 445*x^12 - 333*x^10 + 572*x^8 - 107*x^6 - 50*x^4 + 15*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 5*x^20 - 20*x^18 + 90*x^16 + 144*x^14 - 445*x^12 - 333*x^10 + 572*x^8 - 107*x^6 - 50*x^4 + 15*x^2 - 1)
 

\( x^{22} - 5 x^{20} - 20 x^{18} + 90 x^{16} + 144 x^{14} - 445 x^{12} - 333 x^{10} + 572 x^{8} - 107 x^{6} + \cdots - 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $22$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[14, 4]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(7198079267989980836471065337135104\) \(\medspace = 2^{22}\cdot 23^{20}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(34.59\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(23\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{6439}a^{18}+\frac{2282}{6439}a^{16}-\frac{1694}{6439}a^{14}-\frac{344}{6439}a^{12}-\frac{14}{6439}a^{10}+\frac{272}{6439}a^{8}+\frac{3010}{6439}a^{6}+\frac{1323}{6439}a^{4}+\frac{978}{6439}a^{2}+\frac{2098}{6439}$, $\frac{1}{6439}a^{19}+\frac{2282}{6439}a^{17}-\frac{1694}{6439}a^{15}-\frac{344}{6439}a^{13}-\frac{14}{6439}a^{11}+\frac{272}{6439}a^{9}+\frac{3010}{6439}a^{7}+\frac{1323}{6439}a^{5}+\frac{978}{6439}a^{3}+\frac{2098}{6439}a$, $\frac{1}{895021}a^{20}+\frac{69}{895021}a^{18}-\frac{254705}{895021}a^{16}-\frac{391799}{895021}a^{14}+\frac{278333}{895021}a^{12}-\frac{123282}{895021}a^{10}-\frac{116001}{895021}a^{8}+\frac{410215}{895021}a^{6}+\frac{202533}{895021}a^{4}-\frac{243394}{895021}a^{2}-\frac{355}{895021}$, $\frac{1}{895021}a^{21}+\frac{69}{895021}a^{19}-\frac{254705}{895021}a^{17}-\frac{391799}{895021}a^{15}+\frac{278333}{895021}a^{13}-\frac{123282}{895021}a^{11}-\frac{116001}{895021}a^{9}+\frac{410215}{895021}a^{7}+\frac{202533}{895021}a^{5}-\frac{243394}{895021}a^{3}-\frac{355}{895021}a$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $17$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{1968056}{895021}a^{21}-\frac{9511122}{895021}a^{19}-\frac{40939885}{895021}a^{17}+\frac{170217093}{895021}a^{15}+\frac{311640807}{895021}a^{13}-\frac{822616256}{895021}a^{11}-\frac{791343919}{895021}a^{9}+\frac{988607140}{895021}a^{7}-\frac{49229524}{895021}a^{5}-\frac{101298249}{895021}a^{3}+\frac{12823018}{895021}a$, $a^{21}-5a^{19}-20a^{17}+90a^{15}+144a^{13}-445a^{11}-333a^{9}+572a^{7}-107a^{5}-50a^{3}+15a$, $\frac{3221154}{895021}a^{21}-\frac{15483194}{895021}a^{19}-\frac{67386766}{895021}a^{17}+\frac{276749602}{895021}a^{15}+\frac{516698631}{895021}a^{13}-\frac{1331309826}{895021}a^{11}-\frac{1324653839}{895021}a^{9}+\frac{1576768881}{895021}a^{7}-\frac{55366244}{895021}a^{5}-\frac{163932687}{895021}a^{3}+\frac{20044318}{895021}a$, $\frac{1253098}{895021}a^{20}-\frac{5972072}{895021}a^{18}-\frac{26446881}{895021}a^{16}+\frac{106532509}{895021}a^{14}+\frac{205057824}{895021}a^{12}-\frac{508693570}{895021}a^{10}-\frac{533309920}{895021}a^{8}+\frac{588161741}{895021}a^{6}-\frac{6136720}{895021}a^{4}-\frac{62634438}{895021}a^{2}+\frac{7221300}{895021}$, $\frac{22236}{895021}a^{21}-\frac{133577}{895021}a^{19}-\frac{357002}{895021}a^{17}+\frac{2552150}{895021}a^{15}+\frac{1768138}{895021}a^{13}-\frac{14985078}{895021}a^{11}-\frac{2515322}{895021}a^{9}+\frac{28933580}{895021}a^{7}-\frac{575222}{895021}a^{5}-\frac{7516961}{895021}a^{3}-\frac{1268901}{895021}a$, $\frac{417835}{895021}a^{20}-\frac{2011261}{895021}a^{18}-\frac{8746493}{895021}a^{16}+\frac{36043854}{895021}a^{14}+\frac{67183342}{895021}a^{12}-\frac{174533810}{895021}a^{10}-\frac{173903114}{895021}a^{8}+\frac{210839257}{895021}a^{6}+\frac{450926}{895021}a^{4}-\frac{22608355}{895021}a^{2}+\frac{1319450}{895021}$, $\frac{327519}{895021}a^{21}-\frac{1867135}{895021}a^{19}-\frac{5543508}{895021}a^{17}+\frac{34710048}{895021}a^{15}+\frac{29605589}{895021}a^{13}-\frac{190129236}{895021}a^{11}-\frac{32262023}{895021}a^{9}+\frac{315587066}{895021}a^{7}-\frac{95960107}{895021}a^{5}-\frac{42488861}{895021}a^{3}+\frac{171028}{19043}a$, $\frac{188295}{895021}a^{20}-\frac{1004389}{895021}a^{18}-\frac{3513155}{895021}a^{16}+\frac{18486707}{895021}a^{14}+\frac{22800061}{895021}a^{12}-\frac{97732006}{895021}a^{10}-\frac{45656916}{895021}a^{8}+\frac{149923683}{895021}a^{6}-\frac{25661985}{895021}a^{4}-\frac{19482824}{895021}a^{2}+\frac{2541990}{895021}$, $\frac{1006683}{895021}a^{21}-\frac{4828674}{895021}a^{19}-\frac{21086085}{895021}a^{17}+\frac{86193705}{895021}a^{15}+\frac{161770953}{895021}a^{13}-\frac{413053573}{895021}a^{11}-\frac{412701186}{895021}a^{9}+\frac{483994583}{895021}a^{7}-\frac{29257104}{895021}a^{5}-\frac{53660063}{895021}a^{3}+\frac{9435608}{895021}a$, $\frac{1050799}{895021}a^{21}-\frac{4902440}{895021}a^{19}-\frac{22682463}{895021}a^{17}+\frac{87113053}{895021}a^{15}+\frac{180989645}{895021}a^{13}-\frac{409346833}{895021}a^{11}-\frac{490786604}{895021}a^{9}+\frac{447721548}{895021}a^{7}+\frac{47112945}{895021}a^{5}-\frac{49030238}{895021}a^{3}+\frac{666604}{895021}a$, $\frac{1968056}{895021}a^{21}-\frac{9511122}{895021}a^{19}-\frac{40939885}{895021}a^{17}+\frac{170217093}{895021}a^{15}+\frac{311640807}{895021}a^{13}-\frac{822616256}{895021}a^{11}-\frac{791343919}{895021}a^{9}+\frac{988607140}{895021}a^{7}-\frac{49229524}{895021}a^{5}-\frac{101298249}{895021}a^{3}+\frac{12823018}{895021}a-1$, $\frac{1006683}{895021}a^{21}-\frac{312283}{895021}a^{20}-\frac{4828674}{895021}a^{19}+\frac{1442239}{895021}a^{18}-\frac{21086085}{895021}a^{17}+\frac{6780489}{895021}a^{16}+\frac{86193705}{895021}a^{15}-\frac{25459086}{895021}a^{14}+\frac{161770953}{895021}a^{13}-\frac{54265074}{895021}a^{12}-\frac{413053573}{895021}a^{11}+\frac{117143078}{895021}a^{10}-\frac{412701186}{895021}a^{9}+\frac{144758356}{895021}a^{8}+\frac{483994583}{895021}a^{7}-\frac{118100884}{895021}a^{6}-\frac{29257104}{895021}a^{5}-\frac{97078}{895021}a^{4}-\frac{53660063}{895021}a^{3}+\frac{10848978}{895021}a^{2}+\frac{9435608}{895021}a-\frac{2064803}{895021}$, $\frac{521141}{895021}a^{21}-\frac{2583469}{895021}a^{19}-\frac{10556397}{895021}a^{17}+\frac{46545688}{895021}a^{15}+\frac{77596454}{895021}a^{13}-\frac{230139607}{895021}a^{11}-\frac{188525031}{895021}a^{9}+\frac{295577330}{895021}a^{7}-\frac{26828507}{895021}a^{5}-\frac{26632272}{895021}a^{3}+\frac{300154}{895021}a+1$, $\frac{210531}{895021}a^{21}-\frac{229540}{895021}a^{20}-\frac{1137966}{895021}a^{19}+\frac{1006872}{895021}a^{18}-\frac{3870157}{895021}a^{17}+\frac{5233338}{895021}a^{16}+\frac{21038857}{895021}a^{15}-\frac{17557147}{895021}a^{14}+\frac{24568199}{895021}a^{13}-\frac{44383281}{895021}a^{12}-\frac{112717084}{895021}a^{11}+\frac{76801804}{895021}a^{10}-\frac{48172238}{895021}a^{9}+\frac{128246198}{895021}a^{8}+\frac{178857263}{895021}a^{7}-\frac{60915574}{895021}a^{6}-\frac{26237207}{895021}a^{5}-\frac{26112911}{895021}a^{4}-\frac{26999785}{895021}a^{3}+\frac{3125531}{895021}a^{2}+\frac{46130}{19043}a+\frac{1222540}{895021}$, $\frac{4830972}{895021}a^{21}-\frac{971689}{895021}a^{20}-\frac{23543208}{895021}a^{19}+\frac{101008}{19043}a^{18}-\frac{99586282}{895021}a^{17}+\frac{19969768}{895021}a^{16}+\frac{422104380}{895021}a^{15}-\frac{85137899}{895021}a^{14}+\frac{748842718}{895021}a^{13}-\frac{3181069}{19043}a^{12}-\frac{2053612669}{895021}a^{11}+\frac{414751084}{895021}a^{10}-\frac{1867030565}{895021}a^{9}+\frac{369761910}{895021}a^{8}+\frac{2519903902}{895021}a^{7}-\frac{510827888}{895021}a^{6}-\frac{201206678}{895021}a^{5}+\frac{49289027}{895021}a^{4}-\frac{257361971}{895021}a^{3}+\frac{51307936}{895021}a^{2}+\frac{36929665}{895021}a-\frac{8924945}{895021}$, $\frac{9870515}{895021}a^{21}+\frac{4107371}{895021}a^{20}-\frac{47940815}{895021}a^{19}-\frac{145518}{6533}a^{18}-\frac{4346862}{19043}a^{17}-\frac{85071559}{895021}a^{16}+\frac{859273087}{895021}a^{15}+\frac{357257540}{895021}a^{14}+\frac{1545088063}{895021}a^{13}+\frac{643856165}{895021}a^{12}-\frac{4173917452}{895021}a^{11}-\frac{1734222137}{895021}a^{10}-\frac{3891151467}{895021}a^{9}-\frac{1622282952}{895021}a^{8}+\frac{5099754234}{895021}a^{7}+\frac{2114623989}{895021}a^{6}-\frac{305544380}{895021}a^{5}-\frac{128108889}{895021}a^{4}-\frac{542939073}{895021}a^{3}-\frac{226244901}{895021}a^{2}+\frac{66624473}{895021}a+\frac{27933129}{895021}$, $\frac{784465}{895021}a^{21}-\frac{745354}{895021}a^{20}-\frac{4148499}{895021}a^{19}+\frac{3878396}{895021}a^{18}-\frac{14774461}{895021}a^{17}+\frac{14290001}{895021}a^{16}+\frac{76087709}{895021}a^{15}-\frac{70753902}{895021}a^{14}+\frac{97460659}{895021}a^{13}-\frac{96788931}{895021}a^{12}-\frac{398486317}{895021}a^{11}+\frac{364663046}{895021}a^{10}-\frac{201042995}{895021}a^{9}+\frac{206165137}{895021}a^{8}+\frac{598611487}{895021}a^{7}-\frac{526426323}{895021}a^{6}-\frac{99076930}{895021}a^{5}+\frac{95509181}{895021}a^{4}-\frac{78463021}{895021}a^{3}+\frac{65097216}{895021}a^{2}+\frac{7097097}{895021}a-\frac{9357766}{895021}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 1094333914.95 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 1094333914.95 \cdot 1}{2\cdot\sqrt{7198079267989980836471065337135104}}\cr\approx \mathstrut & 0.164683831790 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^20 - 20*x^18 + 90*x^16 + 144*x^14 - 445*x^12 - 333*x^10 + 572*x^8 - 107*x^6 - 50*x^4 + 15*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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^22 - 5*x^20 - 20*x^18 + 90*x^16 + 144*x^14 - 445*x^12 - 333*x^10 + 572*x^8 - 107*x^6 - 50*x^4 + 15*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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 5*x^20 - 20*x^18 + 90*x^16 + 144*x^14 - 445*x^12 - 333*x^10 + 572*x^8 - 107*x^6 - 50*x^4 + 15*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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 5*x^20 - 20*x^18 + 90*x^16 + 144*x^14 - 445*x^12 - 333*x^10 + 572*x^8 - 107*x^6 - 50*x^4 + 15*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

$C_2^{10}:C_{11}$ (as 22T23):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 11264
The 104 conjugacy class representatives for $C_2^{10}:C_{11}$
Character table for $C_2^{10}:C_{11}$

Intermediate fields

\(\Q(\zeta_{23})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: data not computed
Minimal sibling: 22.10.7198079267989980836471065337135104.11

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.11.0.1}{11} }^{2}$ ${\href{/padicField/5.11.0.1}{11} }^{2}$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.11.0.1}{11} }^{2}$ ${\href{/padicField/19.11.0.1}{11} }^{2}$ R ${\href{/padicField/29.11.0.1}{11} }^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{2}$ ${\href{/padicField/37.11.0.1}{11} }^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ ${\href{/padicField/43.11.0.1}{11} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{10}$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ ${\href{/padicField/59.11.0.1}{11} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.11.2.22a29.1$x^{22} + 2 x^{18} + 2 x^{16} + 2 x^{15} + 2 x^{14} + 4 x^{13} + 2 x^{12} + 2 x^{11} + 2 x^{9} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 5 x^{4} + 4 x^{3} + 4 x^{2} + 2 x + 3$$2$$11$$22$not computednot computed
\(23\) Copy content Toggle raw display 23.1.11.10a1.1$x^{11} + 23$$11$$1$$10$$C_{11}$$$[\ ]_{11}$$
23.1.11.10a1.1$x^{11} + 23$$11$$1$$10$$C_{11}$$$[\ ]_{11}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)