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Properties

Label 18.0.896...000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-8.970\times 10^{45}$
Root discriminant \(357.22\)
Ramified primes $2,3,5,19$
Class number $8748$ (GRH)
Class group [3, 3, 3, 3, 6, 18] (GRH)
Galois group $C_3^2:C_6$ (as 18T23)

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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 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607)
 
Copy content gp:K = bnfinit(y^18 - 60*y^15 - 28551*y^12 + 1705300*y^9 + 313150767*y^6 - 184875000*y^3 + 40353607, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607)
 

\( x^{18} - 60x^{15} - 28551x^{12} + 1705300x^{9} + 313150767x^{6} - 184875000x^{3} + 40353607 \) 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:  $[0, 9]$
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:   \(-8969580780515349163336915861800987000000000000\) \(\medspace = -\,2^{12}\cdot 3^{39}\cdot 5^{12}\cdot 19^{12}\) 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:  \(357.22\)
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:  $2^{2/3}3^{121/54}5^{2/3}19^{2/3}\approx 387.502136220015$
Ramified primes:   \(2\), \(3\), \(5\), \(19\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_3^2$
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.
Maximal CM subfield:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{18}a^{6}-\frac{1}{9}a^{3}+\frac{1}{18}$, $\frac{1}{18}a^{7}-\frac{1}{9}a^{4}+\frac{1}{18}a$, $\frac{1}{54}a^{8}+\frac{1}{54}a^{7}+\frac{1}{54}a^{6}-\frac{1}{27}a^{5}-\frac{1}{27}a^{4}-\frac{1}{27}a^{3}+\frac{1}{54}a^{2}+\frac{1}{54}a+\frac{1}{54}$, $\frac{1}{108}a^{9}-\frac{1}{36}a^{6}+\frac{1}{36}a^{3}-\frac{1}{108}$, $\frac{1}{108}a^{10}-\frac{1}{36}a^{7}+\frac{1}{36}a^{4}-\frac{1}{108}a$, $\frac{1}{324}a^{11}+\frac{1}{324}a^{10}+\frac{1}{324}a^{9}-\frac{1}{108}a^{8}-\frac{1}{108}a^{7}-\frac{1}{108}a^{6}+\frac{1}{108}a^{5}+\frac{1}{108}a^{4}+\frac{1}{108}a^{3}-\frac{1}{324}a^{2}-\frac{1}{324}a-\frac{1}{324}$, $\frac{1}{326916}a^{12}+\frac{779}{326916}a^{9}+\frac{2501}{108972}a^{6}-\frac{33007}{326916}a^{3}+\frac{13471}{81729}$, $\frac{1}{980748}a^{13}-\frac{1}{980748}a^{12}+\frac{779}{980748}a^{10}-\frac{779}{980748}a^{9}-\frac{3553}{326916}a^{7}+\frac{3553}{326916}a^{6}+\frac{112289}{980748}a^{4}-\frac{112289}{980748}a^{3}-\frac{36625}{490374}a+\frac{36625}{490374}$, $\frac{1}{980748}a^{14}-\frac{1}{980748}a^{12}+\frac{779}{980748}a^{11}-\frac{779}{980748}a^{9}+\frac{2501}{326916}a^{8}+\frac{1}{54}a^{7}-\frac{8555}{326916}a^{6}+\frac{75965}{980748}a^{5}-\frac{1}{27}a^{4}-\frac{39641}{980748}a^{3}-\frac{13772}{245187}a^{2}+\frac{1}{54}a+\frac{18463}{490374}$, $\frac{1}{15656247196344}a^{15}-\frac{12003005}{15656247196344}a^{12}-\frac{1228492240}{1957030899543}a^{9}+\frac{47756813059}{1957030899543}a^{6}+\frac{1004126499191}{15656247196344}a^{3}+\frac{2642949232301}{15656247196344}$, $\frac{1}{16\cdots 76}a^{16}-\frac{1}{46968741589032}a^{15}-\frac{7674520445}{16\cdots 76}a^{13}+\frac{59893739}{46968741589032}a^{12}-\frac{1757259871387}{80\cdots 88}a^{10}+\frac{48025017881}{11742185397258}a^{9}-\frac{80524949700677}{80\cdots 88}a^{7}-\frac{114406520681}{11742185397258}a^{6}+\frac{381345295989293}{16\cdots 76}a^{4}-\frac{7368709266323}{46968741589032}a^{3}-\frac{295286695812985}{16\cdots 76}a+\frac{5011378892185}{46968741589032}$, $\frac{1}{55\cdots 68}a^{17}-\frac{1}{46968741589032}a^{15}+\frac{912210698227}{55\cdots 68}a^{14}-\frac{35887729}{46968741589032}a^{12}-\frac{22\cdots 65}{27\cdots 84}a^{11}-\frac{1}{324}a^{10}-\frac{13739471933}{23484370794516}a^{9}-\frac{806314387232885}{27\cdots 84}a^{8}-\frac{1}{108}a^{7}+\frac{64206414617}{23484370794516}a^{6}+\frac{45\cdots 49}{55\cdots 68}a^{5}+\frac{1}{36}a^{4}-\frac{1162980063869}{46968741589032}a^{3}+\frac{24\cdots 07}{55\cdots 68}a^{2}-\frac{5}{324}a-\frac{4353702032249}{46968741589032}$ 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:  $C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{18}$, which has order $8748$ (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:  $C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{18}$, which has order $8748$ (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:  $8$
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:   \( -\frac{1}{73689288} a^{15} + \frac{545}{663203592} a^{12} + \frac{63995}{165800898} a^{9} - \frac{141985}{6140774} a^{6} - \frac{2804676875}{663203592} a^{3} + \frac{1158532651}{663203592} \)  (order $6$) 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:   $\frac{81230204768}{23\cdots 07}a^{17}-\frac{1070863769}{26\cdots 96}a^{16}-\frac{4828997311351}{23\cdots 07}a^{14}+\frac{15978233134}{671261598543249}a^{13}-\frac{23\cdots 27}{23\cdots 07}a^{11}+\frac{15304028781899}{13\cdots 98}a^{10}+\frac{13\cdots 24}{23\cdots 07}a^{8}-\frac{454676951737804}{671261598543249}a^{7}+\frac{25\cdots 97}{23\cdots 07}a^{5}-\frac{33\cdots 45}{26\cdots 96}a^{4}-\frac{32\cdots 39}{23\cdots 07}a^{2}+\frac{29\cdots 32}{671261598543249}a-19$, $\frac{226199756467}{92\cdots 28}a^{17}+\frac{64943750}{671261598543249}a^{16}-\frac{13606658783651}{92\cdots 28}a^{14}-\frac{15767075437}{26\cdots 96}a^{13}-\frac{32\cdots 59}{46\cdots 14}a^{11}-\frac{3703721514155}{13\cdots 98}a^{10}+\frac{19\cdots 37}{46\cdots 14}a^{8}+\frac{111979577183083}{671261598543249}a^{7}+\frac{70\cdots 83}{92\cdots 28}a^{5}+\frac{40\cdots 05}{13\cdots 98}a^{4}-\frac{53\cdots 39}{92\cdots 28}a^{2}-\frac{93\cdots 71}{26\cdots 96}a-19$, $\frac{588702725705}{18\cdots 56}a^{17}+\frac{83784265}{149169244120722}a^{16}-\frac{13}{18422322}a^{15}-\frac{34656158318101}{18\cdots 56}a^{14}-\frac{5013397297}{149169244120722}a^{13}+\frac{7085}{165800898}a^{12}-\frac{21\cdots 95}{23\cdots 07}a^{11}-\frac{1197120980008}{74584622060361}a^{10}+\frac{1663870}{82900449}a^{9}+\frac{12\cdots 89}{23\cdots 07}a^{8}+\frac{71346459917206}{74584622060361}a^{7}-\frac{3691610}{3070387}a^{6}+\frac{18\cdots 75}{18\cdots 56}a^{5}+\frac{26\cdots 65}{149169244120722}a^{4}-\frac{36460799375}{165800898}a^{3}+\frac{10\cdots 77}{18\cdots 56}a^{2}-\frac{15\cdots 99}{149169244120722}a+\frac{11579105605}{165800898}$, $\frac{406265155735}{30\cdots 76}a^{17}+\frac{312034505}{315887811079176}a^{16}+\frac{13}{18422322}a^{15}-\frac{24226302993283}{30\cdots 76}a^{14}-\frac{18634692079}{315887811079176}a^{13}-\frac{7085}{165800898}a^{12}-\frac{967406445130550}{25\cdots 23}a^{11}-\frac{1114178038766}{39485976384897}a^{10}-\frac{1663870}{82900449}a^{9}+\frac{17\cdots 19}{76\cdots 69}a^{8}+\frac{132324273531901}{78971952769794}a^{7}+\frac{3691610}{3070387}a^{6}+\frac{12\cdots 61}{30\cdots 76}a^{5}+\frac{97\cdots 35}{315887811079176}a^{4}+\frac{36460799375}{165800898}a^{3}-\frac{99\cdots 83}{10\cdots 92}a^{2}-\frac{27\cdots 73}{315887811079176}a-\frac{11579105605}{165800898}$, $\frac{17\cdots 99}{90\cdots 88}a^{17}-\frac{28\cdots 01}{16\cdots 76}a^{16}+\frac{142299959642051}{23484370794516}a^{15}+\frac{29\cdots 05}{90\cdots 88}a^{14}-\frac{23\cdots 07}{16\cdots 76}a^{13}+\frac{10\cdots 81}{11742185397258}a^{12}-\frac{16\cdots 23}{22\cdots 22}a^{11}+\frac{14\cdots 07}{40\cdots 94}a^{10}-\frac{57\cdots 49}{23484370794516}a^{9}-\frac{62\cdots 50}{11\cdots 61}a^{8}+\frac{67\cdots 11}{20\cdots 47}a^{7}-\frac{13\cdots 63}{23484370794516}a^{6}+\frac{29\cdots 29}{90\cdots 88}a^{5}-\frac{32\cdots 75}{16\cdots 76}a^{4}+\frac{41\cdots 05}{11742185397258}a^{3}-\frac{64\cdots 73}{90\cdots 88}a^{2}+\frac{70\cdots 35}{16\cdots 76}a-\frac{17\cdots 19}{23484370794516}$, $\frac{12\cdots 09}{55\cdots 68}a^{17}-\frac{17\cdots 07}{16\cdots 76}a^{16}-\frac{45\cdots 91}{46968741589032}a^{15}-\frac{18\cdots 37}{55\cdots 68}a^{14}+\frac{32\cdots 25}{16\cdots 76}a^{13}+\frac{79\cdots 57}{46968741589032}a^{12}-\frac{44\cdots 71}{13\cdots 42}a^{11}+\frac{35\cdots 33}{80\cdots 88}a^{10}+\frac{51\cdots 24}{5871092698629}a^{9}+\frac{46\cdots 11}{69\cdots 21}a^{8}-\frac{20\cdots 49}{80\cdots 88}a^{7}-\frac{15\cdots 35}{5871092698629}a^{6}+\frac{23\cdots 55}{55\cdots 68}a^{5}+\frac{19\cdots 81}{16\cdots 76}a^{4}+\frac{63\cdots 47}{46968741589032}a^{3}-\frac{82\cdots 63}{55\cdots 68}a^{2}+\frac{71\cdots 85}{16\cdots 76}a-\frac{13\cdots 77}{46968741589032}$, $\frac{29\cdots 91}{90\cdots 88}a^{17}+\frac{28\cdots 01}{16\cdots 76}a^{16}+\frac{440074827812447}{46968741589032}a^{15}+\frac{26\cdots 57}{90\cdots 88}a^{14}+\frac{23\cdots 07}{16\cdots 76}a^{13}+\frac{32\cdots 51}{46968741589032}a^{12}-\frac{27\cdots 13}{45\cdots 44}a^{11}-\frac{14\cdots 07}{40\cdots 94}a^{10}-\frac{48\cdots 53}{23484370794516}a^{9}-\frac{26\cdots 95}{45\cdots 44}a^{8}-\frac{67\cdots 11}{20\cdots 47}a^{7}-\frac{44\cdots 01}{23484370794516}a^{6}+\frac{31\cdots 27}{90\cdots 88}a^{5}+\frac{32\cdots 75}{16\cdots 76}a^{4}+\frac{52\cdots 07}{46968741589032}a^{3}-\frac{69\cdots 11}{90\cdots 88}a^{2}-\frac{70\cdots 35}{16\cdots 76}a-\frac{11\cdots 09}{46968741589032}$, $\frac{15\cdots 74}{11\cdots 61}a^{17}-\frac{28\cdots 01}{16\cdots 76}a^{16}+\frac{155474908528345}{46968741589032}a^{15}+\frac{65\cdots 13}{22\cdots 22}a^{14}-\frac{23\cdots 07}{16\cdots 76}a^{13}-\frac{10\cdots 73}{46968741589032}a^{12}+\frac{49\cdots 33}{45\cdots 44}a^{11}+\frac{14\cdots 07}{40\cdots 94}a^{10}-\frac{10\cdots 51}{5871092698629}a^{9}-\frac{19\cdots 95}{45\cdots 44}a^{8}+\frac{67\cdots 11}{20\cdots 47}a^{7}-\frac{15\cdots 69}{11742185397258}a^{6}+\frac{11\cdots 99}{45\cdots 44}a^{5}-\frac{32\cdots 75}{16\cdots 76}a^{4}+\frac{35\cdots 87}{46968741589032}a^{3}-\frac{24\cdots 19}{45\cdots 44}a^{2}+\frac{70\cdots 35}{16\cdots 76}a-\frac{77\cdots 71}{46968741589032}$ 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:  \( 4411949748464.114 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 4411949748464.114 \cdot 8748}{6\cdot\sqrt{8969580780515349163336915861800987000000000000}}\cr\approx \mathstrut & 1.03662295991021 \end{aligned}\] (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 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607) 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 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607, 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 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607); 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 - 60*x^15 - 28551*x^12 + 1705300*x^9 + 313150767*x^6 - 184875000*x^3 + 40353607); 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

$C_3^2:C_6$ (as 18T23):

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 54
The 18 conjugacy class representatives for $C_3^2:C_6$
Character table for $C_3^2:C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.87723.1 x3, 6.0.230859741870000.13, 6.0.230859741870000.12, 6.0.196830000.3, 6.0.23085974187.5

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 siblings: data not computed
Degree 27 sibling: data not computed
Minimal sibling: 18.0.996620086723927684815212873533443000000000000.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ R ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{9}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{9}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.2.3.4a1.1$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 5 x + 1$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
2.2.3.4a1.1$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 5 x + 1$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
2.2.3.4a1.1$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 5 x + 1$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
\(3\) Copy content Toggle raw display 3.1.18.39c2.65$x^{18} + 3 x^{15} + 6 x^{12} + 9 x^{9} + 9 x^{8} + 9 x^{5} + 18 x^{4} + 3$$18$$1$$39$not computednot computed
\(5\) Copy content Toggle raw display 5.2.3.4a1.1$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
5.2.3.4a1.1$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
5.2.3.4a1.1$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
\(19\) Copy content Toggle raw display 19.1.3.2a1.3$x^{3} + 76$$3$$1$$2$$C_3$$$[\ ]_{3}$$
19.1.3.2a1.3$x^{3} + 76$$3$$1$$2$$C_3$$$[\ ]_{3}$$
19.1.3.2a1.3$x^{3} + 76$$3$$1$$2$$C_3$$$[\ ]_{3}$$
19.1.3.2a1.3$x^{3} + 76$$3$$1$$2$$C_3$$$[\ ]_{3}$$
19.1.3.2a1.3$x^{3} + 76$$3$$1$$2$$C_3$$$[\ ]_{3}$$
19.1.3.2a1.3$x^{3} + 76$$3$$1$$2$$C_3$$$[\ ]_{3}$$

Spectrum of ring of integers

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