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Properties

Label 16.8.741...000.1
Degree $16$
Signature $[8, 4]$
Discriminant $7.416\times 10^{19}$
Root discriminant \(17.45\)
Ramified primes $2,5,29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6:D_4$ (as 16T969)

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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^16 - 4*x^15 - 6*x^14 + 32*x^13 + 20*x^12 - 96*x^11 - 76*x^10 + 120*x^9 + 162*x^8 + 4*x^7 - 120*x^6 - 72*x^5 + 28*x^4 - 64*x^3 - 64*x^2 + 4)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 - 6*y^14 + 32*y^13 + 20*y^12 - 96*y^11 - 76*y^10 + 120*y^9 + 162*y^8 + 4*y^7 - 120*y^6 - 72*y^5 + 28*y^4 - 64*y^3 - 64*y^2 + 4, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 6*x^14 + 32*x^13 + 20*x^12 - 96*x^11 - 76*x^10 + 120*x^9 + 162*x^8 + 4*x^7 - 120*x^6 - 72*x^5 + 28*x^4 - 64*x^3 - 64*x^2 + 4);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 - 6*x^14 + 32*x^13 + 20*x^12 - 96*x^11 - 76*x^10 + 120*x^9 + 162*x^8 + 4*x^7 - 120*x^6 - 72*x^5 + 28*x^4 - 64*x^3 - 64*x^2 + 4)
 

\( x^{16} - 4 x^{15} - 6 x^{14} + 32 x^{13} + 20 x^{12} - 96 x^{11} - 76 x^{10} + 120 x^{9} + 162 x^{8} + \cdots + 4 \) 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:  $16$
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:  $[8, 4]$
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:   \(74163788185600000000\) \(\medspace = 2^{28}\cdot 5^{8}\cdot 29^{4}\) 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:  \(17.45\)
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:  $2^{15/8}5^{1/2}29^{1/2}\approx 44.16876366153594$
Ramified primes:   \(2\), \(5\), \(29\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2347124319682}a^{15}+\frac{391387497899}{2347124319682}a^{14}-\frac{10745740897}{1173562159841}a^{13}-\frac{212605451481}{2347124319682}a^{12}+\frac{127985234085}{2347124319682}a^{11}+\frac{41052047148}{1173562159841}a^{10}+\frac{261306470507}{2347124319682}a^{9}+\frac{383940382095}{2347124319682}a^{8}-\frac{73148302465}{1173562159841}a^{7}+\frac{582142336762}{1173562159841}a^{6}+\frac{496419579935}{1173562159841}a^{5}+\frac{196668153327}{1173562159841}a^{4}+\frac{387960699572}{1173562159841}a^{3}-\frac{517505828948}{1173562159841}a^{2}+\frac{55704357363}{1173562159841}a-\frac{13357769744}{1173562159841}$ 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:  $C_{2}$, which has order $2$ (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:  $11$
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:   $\frac{155005523977}{2347124319682}a^{15}-\frac{699284844853}{2347124319682}a^{14}-\frac{295645450702}{1173562159841}a^{13}+\frac{5331056438909}{2347124319682}a^{12}+\frac{250233442575}{1173562159841}a^{11}-\frac{7814135488477}{1173562159841}a^{10}-\frac{2191621923542}{1173562159841}a^{9}+\frac{22095557118973}{2347124319682}a^{8}+\frac{8097824586347}{1173562159841}a^{7}-\frac{4314795628174}{1173562159841}a^{6}-\frac{9248673715816}{1173562159841}a^{5}-\frac{1820756550111}{1173562159841}a^{4}+\frac{3960389510142}{1173562159841}a^{3}-\frac{5608315514724}{1173562159841}a^{2}-\frac{800863853652}{1173562159841}a+\frac{1355425470662}{1173562159841}$, $\frac{365998266175}{2347124319682}a^{15}-\frac{1626160857145}{2347124319682}a^{14}-\frac{756428156812}{1173562159841}a^{13}+\frac{6227923972191}{1173562159841}a^{12}+\frac{1104067250909}{1173562159841}a^{11}-\frac{18281599549869}{1173562159841}a^{10}-\frac{6812769381183}{1173562159841}a^{9}+\frac{25065062353451}{1173562159841}a^{8}+\frac{20833674358140}{1173562159841}a^{7}-\frac{6557028105786}{1173562159841}a^{6}-\frac{19942625724473}{1173562159841}a^{5}-\frac{7369217009740}{1173562159841}a^{4}+\frac{7095341415103}{1173562159841}a^{3}-\frac{15077549681297}{1173562159841}a^{2}-\frac{7305754732692}{1173562159841}a+\frac{1783197276025}{1173562159841}$, $\frac{11224803556}{1173562159841}a^{15}-\frac{75891270143}{2347124319682}a^{14}-\frac{248986099507}{2347124319682}a^{13}+\frac{423873976127}{1173562159841}a^{12}+\frac{1259755493583}{2347124319682}a^{11}-\frac{3468237051849}{2347124319682}a^{10}-\frac{2060288366742}{1173562159841}a^{9}+\frac{5721433785017}{2347124319682}a^{8}+\frac{4116144184245}{1173562159841}a^{7}-\frac{853493988694}{1173562159841}a^{6}-\frac{3276151721472}{1173562159841}a^{5}-\frac{814821369719}{1173562159841}a^{4}+\frac{1167589814146}{1173562159841}a^{3}-\frac{1501044609530}{1173562159841}a^{2}-\frac{4449524008628}{1173562159841}a+\frac{412682005957}{1173562159841}$, $\frac{158439215483}{2347124319682}a^{15}-\frac{265990277347}{1173562159841}a^{14}-\frac{1316405990345}{2347124319682}a^{13}+\frac{2197949507165}{1173562159841}a^{12}+\frac{2927842201999}{1173562159841}a^{11}-\frac{12589928715245}{2347124319682}a^{10}-\frac{9413640768091}{1173562159841}a^{9}+\frac{5108135075599}{1173562159841}a^{8}+\frac{15294254285921}{1173562159841}a^{7}+\frac{7248957142659}{1173562159841}a^{6}-\frac{6364614218243}{1173562159841}a^{5}-\frac{8750853130504}{1173562159841}a^{4}-\frac{442184742355}{1173562159841}a^{3}-\frac{4702043923907}{1173562159841}a^{2}-\frac{7216570975544}{1173562159841}a-\frac{1342309898133}{1173562159841}$, $\frac{154485317205}{1173562159841}a^{15}-\frac{678429719680}{1173562159841}a^{14}-\frac{1317319657679}{2347124319682}a^{13}+\frac{5177465259563}{1173562159841}a^{12}+\frac{2124876717259}{2347124319682}a^{11}-\frac{29953540602425}{2347124319682}a^{10}-\frac{5927992848353}{1173562159841}a^{9}+\frac{19664356623257}{1173562159841}a^{8}+\frac{16907796130165}{1173562159841}a^{7}-\frac{3736091359346}{1173562159841}a^{6}-\frac{14163417078349}{1173562159841}a^{5}-\frac{5460563216059}{1173562159841}a^{4}+\frac{2950800214949}{1173562159841}a^{3}-\frac{14938266425329}{1173562159841}a^{2}-\frac{6088340737861}{1173562159841}a+\frac{1344824310379}{1173562159841}$, $\frac{144593534123}{2347124319682}a^{15}-\frac{319665697211}{1173562159841}a^{14}-\frac{603570961315}{2347124319682}a^{13}+\frac{2452327051940}{1173562159841}a^{12}+\frac{432380429460}{1173562159841}a^{11}-\frac{7216684647266}{1173562159841}a^{10}-\frac{5137805194985}{2347124319682}a^{9}+\frac{10051172154499}{1173562159841}a^{8}+\frac{7922971653674}{1173562159841}a^{7}-\frac{3297453108702}{1173562159841}a^{6}-\frac{7888030027676}{1173562159841}a^{5}-\frac{2576158731584}{1173562159841}a^{4}+\frac{3037681929128}{1173562159841}a^{3}-\frac{5544172795792}{1173562159841}a^{2}-\frac{2232219593449}{1173562159841}a+\frac{1600939184079}{1173562159841}$, $\frac{14733387179}{2347124319682}a^{15}-\frac{88409390019}{2347124319682}a^{14}+\frac{8749787255}{1173562159841}a^{13}+\frac{738157460875}{2347124319682}a^{12}-\frac{714502010815}{2347124319682}a^{11}-\frac{2717265268987}{2347124319682}a^{10}+\frac{2560199207385}{2347124319682}a^{9}+\frac{3237040018189}{1173562159841}a^{8}-\frac{1536961193081}{1173562159841}a^{7}-\frac{5089513621046}{1173562159841}a^{6}-\frac{745413643113}{1173562159841}a^{5}+\frac{4173480499829}{1173562159841}a^{4}+\frac{2677728229421}{1173562159841}a^{3}-\frac{1843382040961}{1173562159841}a^{2}-\frac{669848461401}{1173562159841}a+\frac{1289081347619}{1173562159841}$, $\frac{205141139801}{2347124319682}a^{15}-\frac{443416226135}{1173562159841}a^{14}-\frac{474788029573}{1173562159841}a^{13}+\frac{6873802279007}{2347124319682}a^{12}+\frac{1943033108653}{2347124319682}a^{11}-\frac{20209108765227}{2347124319682}a^{10}-\frac{9513902897027}{2347124319682}a^{9}+\frac{26638690073573}{2347124319682}a^{8}+\frac{13000638647558}{1173562159841}a^{7}-\frac{1906909672576}{1173562159841}a^{6}-\frac{11937449584106}{1173562159841}a^{5}-\frac{6244680324082}{1173562159841}a^{4}+\frac{3344548954219}{1173562159841}a^{3}-\frac{7220204204419}{1173562159841}a^{2}-\frac{3012496356270}{1173562159841}a+\frac{1303908774971}{1173562159841}$, $\frac{274416203546}{1173562159841}a^{15}-\frac{2432338703813}{2347124319682}a^{14}-\frac{1160862938473}{1173562159841}a^{13}+\frac{9414698579325}{1173562159841}a^{12}+\frac{3472176725115}{2347124319682}a^{11}-\frac{56285117630067}{2347124319682}a^{10}-\frac{10010787110999}{1173562159841}a^{9}+\frac{80222919126647}{2347124319682}a^{8}+\frac{31033886701346}{1173562159841}a^{7}-\frac{14687041103244}{1173562159841}a^{6}-\frac{32088640946029}{1173562159841}a^{5}-\frac{7408694152306}{1173562159841}a^{4}+\frac{13220975885458}{1173562159841}a^{3}-\frac{22307131288606}{1173562159841}a^{2}-\frac{10632069098143}{1173562159841}a+\frac{4419740582130}{1173562159841}$, $\frac{5995243177}{2347124319682}a^{15}+\frac{43087536925}{2347124319682}a^{14}-\frac{132948940788}{1173562159841}a^{13}-\frac{110797898956}{1173562159841}a^{12}+\frac{807126497758}{1173562159841}a^{11}+\frac{777291234249}{2347124319682}a^{10}-\frac{3062097574631}{2347124319682}a^{9}-\frac{1307223362397}{1173562159841}a^{8}-\frac{261594789474}{1173562159841}a^{7}+\frac{366663515904}{1173562159841}a^{6}+\frac{2391451826759}{1173562159841}a^{5}+\frac{2985516890114}{1173562159841}a^{4}+\frac{2586329662782}{1173562159841}a^{3}+\frac{272972192978}{1173562159841}a^{2}-\frac{2197337828249}{1173562159841}a-\frac{20182687773}{1173562159841}$, $\frac{158455230821}{2347124319682}a^{15}-\frac{680366852689}{2347124319682}a^{14}-\frac{615597979827}{2347124319682}a^{13}+\frac{4835545519491}{2347124319682}a^{12}+\frac{352904053769}{1173562159841}a^{11}-\frac{12531961210693}{2347124319682}a^{10}-\frac{2112075194657}{1173562159841}a^{9}+\frac{7317074385902}{1173562159841}a^{8}+\frac{5460969159181}{1173562159841}a^{7}-\frac{2000152126478}{1173562159841}a^{6}-\frac{5193034025738}{1173562159841}a^{5}+\frac{536084855624}{1173562159841}a^{4}+\frac{4404617823853}{1173562159841}a^{3}-\frac{5031603631646}{1173562159841}a^{2}+\frac{759126646452}{1173562159841}a+\frac{453555503967}{1173562159841}$ 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:  \( 9015.09092349 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 9015.09092349 \cdot 1}{2\cdot\sqrt{74163788185600000000}}\cr\approx \mathstrut & 0.208835151592 \end{aligned}\] (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^16 - 4*x^15 - 6*x^14 + 32*x^13 + 20*x^12 - 96*x^11 - 76*x^10 + 120*x^9 + 162*x^8 + 4*x^7 - 120*x^6 - 72*x^5 + 28*x^4 - 64*x^3 - 64*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^16 - 4*x^15 - 6*x^14 + 32*x^13 + 20*x^12 - 96*x^11 - 76*x^10 + 120*x^9 + 162*x^8 + 4*x^7 - 120*x^6 - 72*x^5 + 28*x^4 - 64*x^3 - 64*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^16 - 4*x^15 - 6*x^14 + 32*x^13 + 20*x^12 - 96*x^11 - 76*x^10 + 120*x^9 + 162*x^8 + 4*x^7 - 120*x^6 - 72*x^5 + 28*x^4 - 64*x^3 - 64*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^16 - 4*x^15 - 6*x^14 + 32*x^13 + 20*x^12 - 96*x^11 - 76*x^10 + 120*x^9 + 162*x^8 + 4*x^7 - 120*x^6 - 72*x^5 + 28*x^4 - 64*x^3 - 64*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

$C_2^6:D_4$ (as 16T969):

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 512
The 44 conjugacy class representatives for $C_2^6:D_4$
Character table for $C_2^6:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, 8.4.74240000.1, 8.4.74240000.2, 8.4.2152960000.3

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.4.4635236761600000000.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.2.8.28a19.1$x^{16} + 10 x^{15} + 50 x^{14} + 168 x^{13} + 420 x^{12} + 826 x^{11} + 1318 x^{10} + 1742 x^{9} + 1931 x^{8} + 1810 x^{7} + 1438 x^{6} + 966 x^{5} + 542 x^{4} + 248 x^{3} + 88 x^{2} + 22 x + 5$$8$$2$$28$16T171$$[2, 2, 2, 2]^{4}$$
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(29\) Copy content Toggle raw display 29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.2.1a1.1$x^{2} + 29$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.2.1a1.2$x^{2} + 58$$2$$1$$1$$C_2$$$[\ ]_{2}$$
29.1.2.1a1.1$x^{2} + 29$$2$$1$$1$$C_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)