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Properties

Label 18.0.245...008.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.455\times 10^{52}$
Root discriminant \(813.88\)
Ramified primes $2,3,7,11,13$
Class number $17006112$ (GRH)
Class group [3, 6, 18, 18, 54, 54] (GRH)
Galois group $S_3^2$ (as 18T11)

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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 - 24*x^16 - 1944*x^14 + 203652*x^12 - 2350512*x^10 + 27599616*x^8 - 727155792*x^6 + 28255106880*x^4 - 372967410816*x^2 + 2051320759488)
 
Copy content gp:K = bnfinit(y^18 - 24*y^16 - 1944*y^14 + 203652*y^12 - 2350512*y^10 + 27599616*y^8 - 727155792*y^6 + 28255106880*y^4 - 372967410816*y^2 + 2051320759488, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 24*x^16 - 1944*x^14 + 203652*x^12 - 2350512*x^10 + 27599616*x^8 - 727155792*x^6 + 28255106880*x^4 - 372967410816*x^2 + 2051320759488);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 24*x^16 - 1944*x^14 + 203652*x^12 - 2350512*x^10 + 27599616*x^8 - 727155792*x^6 + 28255106880*x^4 - 372967410816*x^2 + 2051320759488)
 

\( x^{18} - 24 x^{16} - 1944 x^{14} + 203652 x^{12} - 2350512 x^{10} + 27599616 x^{8} + \cdots + 2051320759488 \) 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:   \(-24550556916950676185786873153998466351700592821547008\) \(\medspace = -\,2^{12}\cdot 3^{21}\cdot 7^{15}\cdot 11^{9}\cdot 13^{15}\) 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:  \(813.88\)
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:  $2^{2/3}3^{7/6}7^{5/6}11^{1/2}13^{5/6}\approx 813.8773527198483$
Ramified primes:   \(2\), \(3\), \(7\), \(11\), \(13\) 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:  \(\Q(\sqrt{-3003}) \)
$\Aut(K/\Q)$:   $S_3$
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:  \(\Q(\sqrt{-3003}) \)

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

$1$, $a$, $a^{2}$, $\frac{1}{6}a^{3}$, $\frac{1}{6}a^{4}$, $\frac{1}{12}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{72}a^{6}-\frac{1}{12}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{72}a^{7}-\frac{1}{2}a$, $\frac{1}{432}a^{8}-\frac{1}{12}a^{4}-\frac{1}{12}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{432}a^{9}-\frac{1}{12}a^{4}-\frac{1}{12}a^{3}-\frac{1}{2}a$, $\frac{1}{9504}a^{10}-\frac{1}{4752}a^{8}-\frac{1}{144}a^{7}+\frac{1}{264}a^{6}-\frac{19}{264}a^{4}-\frac{31}{132}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{28512}a^{11}-\frac{1}{1188}a^{9}+\frac{1}{792}a^{7}+\frac{25}{792}a^{5}-\frac{1}{44}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6272640}a^{12}+\frac{1}{58080}a^{10}-\frac{547}{522720}a^{8}+\frac{17}{29040}a^{6}-\frac{391}{4840}a^{4}-\frac{1}{12}a^{3}-\frac{491}{1320}a^{2}-\frac{3}{40}$, $\frac{1}{6272640}a^{13}+\frac{1}{58080}a^{11}-\frac{547}{522720}a^{9}+\frac{17}{29040}a^{7}+\frac{37}{14520}a^{5}-\frac{1}{12}a^{4}-\frac{17}{440}a^{3}-\frac{1}{2}a^{2}-\frac{3}{40}a$, $\frac{1}{206997120}a^{14}+\frac{1}{22999680}a^{12}-\frac{19}{1437480}a^{10}+\frac{1213}{5749920}a^{8}-\frac{133}{319440}a^{6}-\frac{119}{7260}a^{4}-\frac{1}{12}a^{3}+\frac{13}{44}a^{2}-\frac{1}{2}a-\frac{11}{40}$, $\frac{1}{827988480}a^{15}+\frac{7}{137998080}a^{13}-\frac{1}{12545280}a^{12}+\frac{23}{22999680}a^{11}-\frac{1}{116160}a^{10}+\frac{101}{106480}a^{9}-\frac{221}{348480}a^{8}+\frac{421}{119790}a^{7}+\frac{1159}{174240}a^{6}-\frac{67}{19360}a^{5}-\frac{1247}{29040}a^{4}-\frac{107}{1760}a^{3}+\frac{347}{880}a^{2}-\frac{37}{80}a+\frac{3}{80}$, $\frac{1}{19\cdots 40}a^{16}+\frac{847007543069}{10\cdots 80}a^{14}-\frac{1}{12545280}a^{13}-\frac{10943641908217}{17\cdots 80}a^{12}-\frac{1}{116160}a^{11}-\frac{41\cdots 49}{13\cdots 60}a^{10}-\frac{221}{348480}a^{9}+\frac{12\cdots 67}{22\cdots 60}a^{8}-\frac{17}{58080}a^{7}+\frac{58\cdots 91}{13\cdots 40}a^{6}+\frac{391}{9680}a^{5}-\frac{534554874302837}{12\cdots 40}a^{4}-\frac{59}{2640}a^{3}+\frac{255764997259391}{18\cdots 40}a^{2}+\frac{23}{80}a+\frac{2719180801851}{14147594929820}$, $\frac{1}{63\cdots 20}a^{17}-\frac{127197967335403}{21\cdots 40}a^{15}-\frac{1}{413994240}a^{14}+\frac{246758948570933}{70\cdots 28}a^{13}-\frac{1}{45999360}a^{12}-\frac{37\cdots 89}{35\cdots 64}a^{11}+\frac{19}{2874960}a^{10}+\frac{24\cdots 05}{98\cdots 24}a^{9}+\frac{12097}{11499840}a^{8}+\frac{33\cdots 53}{89\cdots 84}a^{7}+\frac{133}{638880}a^{6}+\frac{34\cdots 78}{12\cdots 35}a^{5}-\frac{81}{2420}a^{4}-\frac{19\cdots 83}{12\cdots 40}a^{3}+\frac{19}{132}a^{2}-\frac{55688318621483}{373496506147248}a+\frac{31}{80}$ 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:  $C_{3}\times C_{6}\times C_{18}\times C_{18}\times C_{54}\times C_{54}$, which has order $17006112$ (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:  $C_{3}\times C_{6}\times C_{18}\times C_{18}\times C_{54}\times C_{54}$, which has order $17006112$ (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:   \( -1 \)  (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:   $\frac{1072548565843}{63\cdots 20}a^{17}-\frac{202480987}{16\cdots 92}a^{16}+\frac{16344613964551}{26\cdots 80}a^{15}+\frac{143683987}{53\cdots 64}a^{14}-\frac{78107057490497}{88\cdots 60}a^{13}+\frac{187453516}{695026094729373}a^{12}-\frac{17\cdots 47}{17\cdots 32}a^{11}-\frac{882909226577}{44\cdots 72}a^{10}+\frac{13\cdots 75}{12\cdots 28}a^{9}-\frac{418171477213}{24\cdots 04}a^{8}+\frac{20\cdots 21}{89\cdots 84}a^{7}-\frac{64326391991}{14040931206654}a^{6}+\frac{96\cdots 03}{40\cdots 20}a^{5}+\frac{18631670239}{1276448291514}a^{4}-\frac{27\cdots 57}{77\cdots 90}a^{3}-\frac{455231558327}{309442010064}a^{2}+\frac{11\cdots 91}{933741265368120}a-\frac{24644484157}{1172128826}$, $\frac{3269089479119}{37\cdots 60}a^{17}-\frac{1661206477}{80\cdots 60}a^{16}-\frac{2555874194689}{15\cdots 40}a^{15}+\frac{11868285287}{26\cdots 20}a^{14}-\frac{9532100194915}{52\cdots 48}a^{13}+\frac{12259934407}{27\cdots 20}a^{12}+\frac{17\cdots 21}{10\cdots 96}a^{11}-\frac{9231612028121}{22\cdots 60}a^{10}-\frac{302434086096977}{28\cdots 36}a^{9}+\frac{11967236593333}{37\cdots 60}a^{8}+\frac{551381472502633}{52\cdots 52}a^{7}-\frac{8510057717}{280818624133080}a^{6}-\frac{16\cdots 81}{23\cdots 60}a^{5}+\frac{724567509029}{5105793166056}a^{4}+\frac{20\cdots 91}{906278286974940}a^{3}-\frac{538243796181}{103147336688}a^{2}-\frac{248726907126971}{10985191357272}a+\frac{195164637156}{2930322065}$, $\frac{913966115}{14\cdots 92}a^{16}-\frac{30366796993}{24\cdots 20}a^{14}-\frac{2971338491713}{24\cdots 20}a^{12}+\frac{1239668347717}{10\cdots 68}a^{10}-\frac{14632333099019}{13\cdots 24}a^{8}+\frac{12831439124833}{61\cdots 92}a^{6}-\frac{908412335321}{31124708845604}a^{4}+\frac{501680465553}{321536248405}a^{2}-\frac{32344570319713}{1286144993620}$, $\frac{1969802575507}{10\cdots 80}a^{16}-\frac{6895528449479}{17\cdots 80}a^{14}-\frac{44980705167989}{11\cdots 80}a^{12}+\frac{81\cdots 87}{22\cdots 60}a^{10}-\frac{18\cdots 77}{74\cdots 20}a^{8}-\frac{972348436912881}{75\cdots 80}a^{6}-\frac{59\cdots 81}{41\cdots 28}a^{4}+\frac{92\cdots 01}{311247088456040}a^{2}-\frac{25\cdots 93}{14147594929820}$, $\frac{4958858493965}{70\cdots 28}a^{17}+\frac{4252683784027}{21\cdots 60}a^{16}-\frac{30144042297473}{23\cdots 76}a^{15}-\frac{2584409749231}{71\cdots 72}a^{14}-\frac{19\cdots 17}{13\cdots 20}a^{13}-\frac{75393991744711}{17\cdots 68}a^{12}+\frac{66\cdots 29}{49\cdots 20}a^{11}+\frac{17\cdots 87}{44\cdots 20}a^{10}-\frac{91\cdots 21}{10\cdots 60}a^{9}-\frac{31\cdots 67}{14\cdots 40}a^{8}+\frac{20\cdots 73}{16\cdots 60}a^{7}+\frac{20\cdots 19}{15\cdots 60}a^{6}-\frac{10\cdots 71}{22\cdots 40}a^{5}-\frac{93\cdots 73}{82\cdots 56}a^{4}+\frac{66\cdots 56}{427964746627055}a^{3}+\frac{27\cdots 93}{622494176912080}a^{2}-\frac{85\cdots 41}{622494176912080}a-\frac{12\cdots 27}{28295189859640}$, $\frac{102992673075995}{38\cdots 88}a^{17}+\frac{187076098829}{17\cdots 88}a^{16}-\frac{700146566472113}{32\cdots 40}a^{15}-\frac{257590341297167}{18\cdots 20}a^{14}-\frac{32\cdots 57}{53\cdots 40}a^{13}-\frac{15\cdots 29}{63\cdots 24}a^{12}+\frac{60\cdots 27}{13\cdots 60}a^{11}+\frac{29\cdots 13}{15\cdots 60}a^{10}+\frac{43\cdots 49}{22\cdots 60}a^{9}+\frac{14\cdots 79}{65\cdots 40}a^{8}+\frac{37\cdots 57}{13\cdots 40}a^{7}-\frac{59903351078331}{22\cdots 20}a^{6}-\frac{28\cdots 11}{12\cdots 40}a^{5}-\frac{37\cdots 61}{36\cdots 60}a^{4}+\frac{11\cdots 47}{373496506147248}a^{3}+\frac{58\cdots 63}{36617304524240}a^{2}-\frac{17\cdots 43}{14147594929820}a-\frac{11\cdots 23}{1664422932920}$, $\frac{554540843209921}{96\cdots 72}a^{16}-\frac{74\cdots 49}{80\cdots 60}a^{14}-\frac{64\cdots 23}{53\cdots 40}a^{12}+\frac{14\cdots 81}{13\cdots 60}a^{10}-\frac{20\cdots 81}{44\cdots 20}a^{8}+\frac{70\cdots 97}{75\cdots 80}a^{6}-\frac{55\cdots 71}{15\cdots 80}a^{4}+\frac{40\cdots 03}{311247088456040}a^{2}-\frac{58\cdots 87}{5659037971928}$, $\frac{10\cdots 71}{31\cdots 80}a^{17}+\frac{12\cdots 27}{65\cdots 20}a^{16}+\frac{18\cdots 31}{20\cdots 20}a^{15}-\frac{42\cdots 51}{32\cdots 60}a^{14}-\frac{50\cdots 69}{69\cdots 40}a^{13}-\frac{13\cdots 47}{32\cdots 76}a^{12}+\frac{12\cdots 71}{27\cdots 65}a^{11}+\frac{13\cdots 57}{40\cdots 20}a^{10}+\frac{43\cdots 91}{57\cdots 20}a^{9}+\frac{33\cdots 83}{22\cdots 40}a^{8}+\frac{89\cdots 87}{87\cdots 20}a^{7}+\frac{18\cdots 77}{41\cdots 80}a^{6}-\frac{11\cdots 03}{19\cdots 80}a^{5}-\frac{29\cdots 43}{37\cdots 80}a^{4}+\frac{29\cdots 17}{604185524649960}a^{3}+\frac{41\cdots 99}{11318075943856}a^{2}+\frac{20\cdots 83}{36617304524240}a-\frac{46\cdots 34}{321536248405}$ 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:  \( 2730395051431.8765 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 2730395051431.8765 \cdot 17006112}{2\cdot\sqrt{24550556916950676185786873153998466351700592821547008}}\cr\approx \mathstrut & 2.26145873999205 \end{aligned}\] (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 - 24*x^16 - 1944*x^14 + 203652*x^12 - 2350512*x^10 + 27599616*x^8 - 727155792*x^6 + 28255106880*x^4 - 372967410816*x^2 + 2051320759488) 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 - 24*x^16 - 1944*x^14 + 203652*x^12 - 2350512*x^10 + 27599616*x^8 - 727155792*x^6 + 28255106880*x^4 - 372967410816*x^2 + 2051320759488, 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 - 24*x^16 - 1944*x^14 + 203652*x^12 - 2350512*x^10 + 27599616*x^8 - 727155792*x^6 + 28255106880*x^4 - 372967410816*x^2 + 2051320759488); 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 - 24*x^16 - 1944*x^14 + 203652*x^12 - 2350512*x^10 + 27599616*x^8 - 727155792*x^6 + 28255106880*x^4 - 372967410816*x^2 + 2051320759488); 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

$S_3^2$ (as 18T11):

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 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3003}) \), 3.1.108108.1 x3, 3.1.24843.1, 6.0.224258431984587.4, 6.0.35097081010992.2, 9.1.259932343846602632901696.1 x3

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

Galois closure: data not computed
Degree 6 sibling: 6.2.96879642617341584.2
Degree 9 sibling: 9.1.259932343846602632901696.1
Degree 12 sibling: deg 12
Degree 18 siblings: 18.0.202694470132765384910848433830619515622399029248.2, deg 18
Minimal sibling: 6.2.96879642617341584.2

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 ${\href{/padicField/5.2.0.1}{2} }^{9}$ R R R ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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

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 2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.6.7a1.3$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$$[\frac{3}{2}]_{2}$$
3.2.6.14a2.1$x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3126 x^{4} + 2264 x^{3} + 1200 x^{2} + 432 x + 91$$6$$2$$14$$D_6$$$[\frac{3}{2}]_{2}^{2}$$
\(7\) Copy content Toggle raw display 7.1.6.5a1.1$x^{6} + 7$$6$$1$$5$$C_6$$$[\ ]_{6}$$
7.1.6.5a1.1$x^{6} + 7$$6$$1$$5$$C_6$$$[\ ]_{6}$$
7.1.6.5a1.1$x^{6} + 7$$6$$1$$5$$C_6$$$[\ ]_{6}$$
\(11\) Copy content Toggle raw display 11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(13\) Copy content Toggle raw display 13.1.6.5a1.4$x^{6} + 52$$6$$1$$5$$C_6$$$[\ ]_{6}$$
13.1.6.5a1.4$x^{6} + 52$$6$$1$$5$$C_6$$$[\ ]_{6}$$
13.1.6.5a1.4$x^{6} + 52$$6$$1$$5$$C_6$$$[\ ]_{6}$$

Spectrum of ring of integers

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