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Properties

Label 18.18.404...616.1
Degree $18$
Signature $[18, 0]$
Discriminant $4.043\times 10^{32}$
Root discriminant \(64.79\)
Ramified primes $2,13,229$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2:A_4^2.S_4$ (as 18T588)

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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^18 - 30*x^16 + 298*x^14 - 1453*x^12 + 3945*x^10 - 6162*x^8 + 5379*x^6 - 2364*x^4 + 397*x^2 - 4)
 
Copy content gp:K = bnfinit(y^18 - 30*y^16 + 298*y^14 - 1453*y^12 + 3945*y^10 - 6162*y^8 + 5379*y^6 - 2364*y^4 + 397*y^2 - 4, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 30*x^16 + 298*x^14 - 1453*x^12 + 3945*x^10 - 6162*x^8 + 5379*x^6 - 2364*x^4 + 397*x^2 - 4);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 30*x^16 + 298*x^14 - 1453*x^12 + 3945*x^10 - 6162*x^8 + 5379*x^6 - 2364*x^4 + 397*x^2 - 4)
 

\( x^{18} - 30x^{16} + 298x^{14} - 1453x^{12} + 3945x^{10} - 6162x^{8} + 5379x^{6} - 2364x^{4} + 397x^{2} - 4 \) 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:  $18$
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:  $[18, 0]$
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:   \(404306392283083098720059869167616\) \(\medspace = 2^{16}\cdot 13^{8}\cdot 229^{8}\) 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:  \(64.79\)
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:   \(2\), \(13\), \(229\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{59119}a^{16}-\frac{8813}{59119}a^{14}+\frac{18106}{59119}a^{12}+\frac{3659}{59119}a^{10}+\frac{27684}{59119}a^{8}+\frac{1713}{59119}a^{6}-\frac{23674}{59119}a^{4}+\frac{4855}{59119}a^{2}-\frac{16269}{59119}$, $\frac{1}{236476}a^{17}-\frac{1}{118238}a^{16}+\frac{25153}{118238}a^{15}-\frac{25153}{59119}a^{14}+\frac{9053}{118238}a^{13}-\frac{9053}{59119}a^{12}+\frac{3659}{236476}a^{11}-\frac{3659}{118238}a^{10}-\frac{31435}{236476}a^{9}+\frac{31435}{118238}a^{8}-\frac{28703}{118238}a^{7}+\frac{28703}{59119}a^{6}-\frac{82793}{236476}a^{5}-\frac{35445}{118238}a^{4}-\frac{13566}{59119}a^{3}+\frac{27132}{59119}a^{2}+\frac{101969}{236476}a+\frac{16269}{118238}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $C_{2}\times C_{2}$, which has order $4$ (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:   $\frac{8969}{59119}a^{16}-\frac{238170}{59119}a^{14}+\frac{1825510}{59119}a^{12}-\frac{6023612}{59119}a^{10}+\frac{8924965}{59119}a^{8}-\frac{4381849}{59119}a^{6}-\frac{1513752}{59119}a^{4}+\frac{1392648}{59119}a^{2}-\frac{247445}{59119}$, $\frac{1405}{59119}a^{16}-\frac{26394}{59119}a^{14}+\frac{17760}{59119}a^{12}+\frac{766089}{59119}a^{10}-\frac{3019351}{59119}a^{8}+\frac{4239454}{59119}a^{6}-\frac{2401852}{59119}a^{4}+\frac{672899}{59119}a^{2}-\frac{38011}{59119}$, $\frac{44313}{118238}a^{17}-\frac{615927}{59119}a^{15}+\frac{5245646}{59119}a^{13}-\frac{41228093}{118238}a^{11}+\frac{83576989}{118238}a^{9}-\frac{43659695}{59119}a^{7}+\frac{42448135}{118238}a^{5}-\frac{3544353}{59119}a^{3}+\frac{205365}{118238}a$, $\frac{4455}{59119}a^{16}-\frac{125137}{59119}a^{14}+\frac{1088056}{59119}a^{12}-\frac{4390805}{59119}a^{10}+\frac{9291669}{59119}a^{8}-\frac{10636356}{59119}a^{6}+\frac{6385478}{59119}a^{4}-\frac{1604742}{59119}a^{2}+\frac{119737}{59119}$, $\frac{20265}{118238}a^{17}-\frac{294068}{59119}a^{15}+\frac{2732262}{59119}a^{13}-\frac{24106143}{118238}a^{11}+\frac{56199119}{118238}a^{9}-\frac{34165236}{59119}a^{7}+\frac{37242045}{118238}a^{5}-\frac{2447159}{59119}a^{3}-\frac{43741}{118238}a$, $\frac{45797}{118238}a^{17}-\frac{652083}{59119}a^{15}+\frac{5851475}{59119}a^{13}-\frac{49868459}{118238}a^{11}+\frac{114373339}{118238}a^{9}-\frac{72657577}{59119}a^{7}+\frac{99186845}{118238}a^{5}-\frac{16554382}{59119}a^{3}+\frac{4202813}{118238}a$, $\frac{9366}{59119}a^{16}-\frac{248910}{59119}a^{14}+\frac{1919312}{59119}a^{12}-\frac{6521916}{59119}a^{10}+\frac{10692949}{59119}a^{8}-\frac{8076594}{59119}a^{6}+\frac{2330326}{59119}a^{4}-\frac{167938}{59119}a^{2}-\frac{25791}{59119}$, $\frac{11541}{59119}a^{16}-\frac{321748}{59119}a^{14}+\frac{2754274}{59119}a^{12}-\frac{10860343}{59119}a^{10}+\frac{21955117}{59119}a^{8}-\frac{22677709}{59119}a^{6}+\frac{11199875}{59119}a^{4}-\frac{2614493}{59119}a^{2}+\frac{119653}{59119}$, $\frac{10969}{236476}a^{17}-\frac{53341}{118238}a^{16}-\frac{182473}{118238}a^{15}+\frac{728960}{59119}a^{14}+\frac{2110721}{118238}a^{13}-\frac{5983100}{59119}a^{12}-\frac{23239997}{236476}a^{11}+\frac{44493707}{118238}a^{10}+\frac{67367153}{236476}a^{9}-\frac{82843581}{118238}a^{8}-\frac{51645419}{118238}a^{7}+\frac{37405188}{59119}a^{6}+\frac{76529647}{236476}a^{5}-\frac{26649675}{118238}a^{4}-\frac{4968927}{59119}a^{3}+\frac{546963}{59119}a^{2}-\frac{979423}{236476}a+\frac{56047}{118238}$, $\frac{87547}{236476}a^{17}-\frac{63955}{118238}a^{16}-\frac{1177201}{118238}a^{15}+\frac{854660}{59119}a^{14}+\frac{9354479}{118238}a^{13}-\frac{6712695}{59119}a^{12}-\frac{67249691}{236476}a^{11}+\frac{47986285}{118238}a^{10}+\frac{120434027}{236476}a^{9}-\frac{87944361}{118238}a^{8}-\frac{49845763}{118238}a^{7}+\frac{41556953}{59119}a^{6}+\frac{20520517}{236476}a^{5}-\frac{37389247}{118238}a^{4}+\frac{3171415}{59119}a^{3}+\frac{3158598}{59119}a^{2}-\frac{3199621}{236476}a+\frac{107733}{118238}$, $\frac{75783}{118238}a^{17}-\frac{47578}{59119}a^{16}-\frac{1068260}{59119}a^{15}+\frac{1333584}{59119}a^{14}+\frac{9387425}{59119}a^{13}-\frac{11552524}{59119}a^{12}-\frac{77660479}{118238}a^{11}+\frac{46425968}{59119}a^{10}+\frac{171347481}{118238}a^{9}-\frac{96696716}{59119}a^{8}-\frac{104142641}{59119}a^{7}+\frac{104309903}{59119}a^{6}+\frac{137508345}{118238}a^{5}-\frac{52826002}{59119}a^{4}-\frac{23219303}{59119}a^{3}+\frac{9091069}{59119}a^{2}+\frac{6338851}{118238}a+\frac{178772}{59119}$, $\frac{53949}{236476}a^{17}-\frac{333}{118238}a^{16}-\frac{747757}{118238}a^{15}+\frac{18949}{59119}a^{14}+\frac{6343971}{118238}a^{13}-\frac{413413}{59119}a^{12}-\frac{49954505}{236476}a^{11}+\frac{5875833}{118238}a^{10}+\frac{102040213}{236476}a^{9}-\frac{18973407}{118238}a^{8}-\frac{53260399}{118238}a^{7}+\frac{15174404}{59119}a^{6}+\frac{46308427}{236476}a^{5}-\frac{22799319}{118238}a^{4}+\frac{612276}{59119}a^{3}+\frac{2945699}{59119}a^{2}-\frac{5927507}{236476}a+\frac{215105}{118238}$, $\frac{4514}{59119}a^{17}+\frac{4455}{59119}a^{16}-\frac{113033}{59119}a^{15}-\frac{125137}{59119}a^{14}+\frac{737454}{59119}a^{13}+\frac{1088056}{59119}a^{12}-\frac{1632807}{59119}a^{11}-\frac{4390805}{59119}a^{10}-\frac{366704}{59119}a^{9}+\frac{9291669}{59119}a^{8}+\frac{6254507}{59119}a^{7}-\frac{10636356}{59119}a^{6}-\frac{7899230}{59119}a^{5}+\frac{6385478}{59119}a^{4}+\frac{2997390}{59119}a^{3}-\frac{1663861}{59119}a^{2}-\frac{130706}{59119}a+\frac{1499}{59119}$, $\frac{28949}{118238}a^{17}-\frac{7682}{59119}a^{16}-\frac{428359}{59119}a^{15}+\frac{187568}{59119}a^{14}+\frac{4139100}{59119}a^{13}-\frac{1106546}{59119}a^{12}-\frac{38680683}{118238}a^{11}+\frac{1273705}{59119}a^{10}+\frac{98321849}{118238}a^{9}+\frac{7372430}{59119}a^{8}-\frac{68820118}{59119}a^{7}-\frac{25160423}{59119}a^{6}+\frac{97810767}{118238}a^{5}+\frac{27681316}{59119}a^{4}-\frac{13409247}{59119}a^{3}-\frac{9864894}{59119}a^{2}+\frac{443625}{118238}a+\frac{119130}{59119}$, $\frac{37251}{236476}a^{17}-\frac{26667}{118238}a^{16}-\frac{534699}{118238}a^{15}+\frac{363837}{59119}a^{14}+\frac{4866285}{118238}a^{13}-\frac{2989424}{59119}a^{12}-\frac{42001495}{236476}a^{11}+\frac{22673493}{118238}a^{10}+\frac{97708555}{236476}a^{9}-\frac{44606001}{118238}a^{8}-\frac{63601263}{118238}a^{7}+\frac{22947380}{59119}a^{6}+\frac{90804733}{236476}a^{5}-\frac{23073653}{118238}a^{4}-\frac{7860681}{59119}a^{3}+\frac{2336363}{59119}a^{2}+\frac{3007419}{236476}a+\frac{30201}{118238}$, $\frac{231}{236476}a^{17}+\frac{24167}{118238}a^{16}+\frac{16681}{118238}a^{15}-\frac{343721}{59119}a^{14}-\frac{509993}{118238}a^{13}+\frac{3058620}{59119}a^{12}+\frac{8412461}{236476}a^{11}-\frac{25199665}{118238}a^{10}-\frac{31145561}{236476}a^{9}+\frac{53314843}{118238}a^{8}+\frac{29550435}{118238}a^{7}-\frac{28754008}{59119}a^{6}-\frac{58380199}{236476}a^{5}+\frac{28578601}{118238}a^{4}+\frac{6798246}{59119}a^{3}-\frac{2315856}{59119}a^{2}-\frac{4349329}{236476}a-\frac{149811}{118238}$, $\frac{42029}{236476}a^{17}-\frac{29175}{118238}a^{16}-\frac{601711}{118238}a^{15}+\frac{419205}{59119}a^{14}+\frac{5437601}{118238}a^{13}-\frac{3820318}{59119}a^{12}-\frac{46038109}{236476}a^{11}+\frac{32887753}{118238}a^{10}+\frac{100748573}{236476}a^{9}-\frac{74782457}{118238}a^{8}-\frac{54365553}{118238}a^{7}+\frac{44801592}{59119}a^{6}+\frac{40001787}{236476}a^{5}-\frac{50013001}{118238}a^{4}+\frac{3052410}{59119}a^{3}+\frac{4051901}{59119}a^{2}-\frac{6147823}{236476}a+\frac{277219}{118238}$ 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:  \( 72756672849.4 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 72756672849.4 \cdot 1}{2\cdot\sqrt{404306392283083098720059869167616}}\cr\approx \mathstrut & 0.474271964426 \end{aligned}\] (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^18 - 30*x^16 + 298*x^14 - 1453*x^12 + 3945*x^10 - 6162*x^8 + 5379*x^6 - 2364*x^4 + 397*x^2 - 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 30*x^16 + 298*x^14 - 1453*x^12 + 3945*x^10 - 6162*x^8 + 5379*x^6 - 2364*x^4 + 397*x^2 - 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 30*x^16 + 298*x^14 - 1453*x^12 + 3945*x^10 - 6162*x^8 + 5379*x^6 - 2364*x^4 + 397*x^2 - 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 30*x^16 + 298*x^14 - 1453*x^12 + 3945*x^10 - 6162*x^8 + 5379*x^6 - 2364*x^4 + 397*x^2 - 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

$C_2^2:A_4^2.S_4$ (as 18T588):

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 13824
The 44 conjugacy class representatives for $C_2^2:A_4^2.S_4$
Character table for $C_2^2:A_4^2.S_4$

Intermediate fields

3.3.229.1, 9.9.78544420275841.1

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 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.9.0.1}{9} }^{2}$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.9.0.1}{9} }^{2}$ R ${\href{/padicField/17.9.0.1}{9} }^{2}$ ${\href{/padicField/19.9.0.1}{9} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.2.4a2.1$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 3$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
2.6.2.12a8.1$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 3$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
13.4.1.0a1.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
13.4.3.8a1.3$x^{12} + 9 x^{10} + 36 x^{9} + 33 x^{8} + 216 x^{7} + 495 x^{6} + 468 x^{5} + 1362 x^{4} + 2160 x^{3} + 900 x^{2} + 144 x + 21$$3$$4$$8$$C_{12}$$$[\ ]_{3}^{4}$$
\(229\) Copy content Toggle raw display $\Q_{229}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{229}$$x$$1$$1$$0$Trivial$$[\ ]$$
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$

Spectrum of ring of integers

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