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Properties

Label 16.10.298...875.1
Degree $16$
Signature $[10, 3]$
Discriminant $-2.989\times 10^{20}$
Root discriminant \(19.04\)
Ramified primes $3,5,179,1021$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^4.C_2\wr C_4$ (as 16T1771)

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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 - x^15 - 13*x^14 + 65*x^12 + 56*x^11 - 119*x^10 - 220*x^9 - 21*x^8 + 220*x^7 + 179*x^6 + 48*x^5 + 30*x^4 - 5*x^3 - 32*x^2 - 8*x + 1)
 
Copy content gp:K = bnfinit(y^16 - y^15 - 13*y^14 + 65*y^12 + 56*y^11 - 119*y^10 - 220*y^9 - 21*y^8 + 220*y^7 + 179*y^6 + 48*y^5 + 30*y^4 - 5*y^3 - 32*y^2 - 8*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 13*x^14 + 65*x^12 + 56*x^11 - 119*x^10 - 220*x^9 - 21*x^8 + 220*x^7 + 179*x^6 + 48*x^5 + 30*x^4 - 5*x^3 - 32*x^2 - 8*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 13*x^14 + 65*x^12 + 56*x^11 - 119*x^10 - 220*x^9 - 21*x^8 + 220*x^7 + 179*x^6 + 48*x^5 + 30*x^4 - 5*x^3 - 32*x^2 - 8*x + 1)
 

\( x^{16} - x^{15} - 13 x^{14} + 65 x^{12} + 56 x^{11} - 119 x^{10} - 220 x^{9} - 21 x^{8} + 220 x^{7} + \cdots + 1 \) 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:  $16$
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:  $[10, 3]$
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:   \(-298892216010498046875\) \(\medspace = -\,3^{8}\cdot 5^{12}\cdot 179\cdot 1021^{2}\) 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:  \(19.04\)
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:  $3^{1/2}5^{3/4}179^{1/2}1021^{1/2}\approx 2475.868173489506$
Ramified primes:   \(3\), \(5\), \(179\), \(1021\) 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:  \(\Q(\sqrt{-179}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{394631354879}a^{15}-\frac{19319158819}{394631354879}a^{14}+\frac{911192251}{394631354879}a^{13}-\frac{160848351330}{394631354879}a^{12}-\frac{52352011165}{394631354879}a^{11}-\frac{81669662009}{394631354879}a^{10}+\frac{171058210141}{394631354879}a^{9}-\frac{99374824570}{394631354879}a^{8}-\frac{4749546609}{394631354879}a^{7}-\frac{140452998914}{394631354879}a^{6}-\frac{167664002266}{394631354879}a^{5}-\frac{186886469353}{394631354879}a^{4}+\frac{190703171777}{394631354879}a^{3}-\frac{134970060906}{394631354879}a^{2}-\frac{157636789665}{394631354879}a+\frac{191269963426}{394631354879}$ 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:  $12$
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:   $\frac{5437526}{188009221}a^{15}+\frac{760857}{188009221}a^{14}-\frac{74409925}{188009221}a^{13}-\frac{82475854}{188009221}a^{12}+\frac{318524804}{188009221}a^{11}+\frac{710426021}{188009221}a^{10}-\frac{126806993}{188009221}a^{9}-\frac{1842367869}{188009221}a^{8}-\frac{1817148094}{188009221}a^{7}+\frac{711756923}{188009221}a^{6}+\frac{2504917100}{188009221}a^{5}+\frac{1722043246}{188009221}a^{4}+\frac{469721899}{188009221}a^{3}+\frac{63803831}{188009221}a^{2}+\frac{5032740}{188009221}a-\frac{117684647}{188009221}$, $\frac{2792126678}{394631354879}a^{15}+\frac{13609057172}{394631354879}a^{14}-\frac{79588293203}{394631354879}a^{13}-\frac{161743571316}{394631354879}a^{12}+\frac{504036088838}{394631354879}a^{11}+\frac{884328432145}{394631354879}a^{10}-\frac{1056131799710}{394631354879}a^{9}-\frac{2350953537558}{394631354879}a^{8}+\frac{294405111933}{394631354879}a^{7}+\frac{2684924680080}{394631354879}a^{6}+\frac{450012514967}{394631354879}a^{5}-\frac{1542207152319}{394631354879}a^{4}+\frac{1145986616984}{394631354879}a^{3}+\frac{1317552468740}{394631354879}a^{2}-\frac{678575896472}{394631354879}a+\frac{229746459537}{394631354879}$, $\frac{18858516240}{394631354879}a^{15}+\frac{15697117053}{394631354879}a^{14}-\frac{325974413437}{394631354879}a^{13}-\frac{342644630870}{394631354879}a^{12}+\frac{1696814585567}{394631354879}a^{11}+\frac{2688614010137}{394631354879}a^{10}-\frac{2616141098314}{394631354879}a^{9}-\frac{7959007169409}{394631354879}a^{8}-\frac{2562479130839}{394631354879}a^{7}+\frac{7304680879011}{394631354879}a^{6}+\frac{6982735280977}{394631354879}a^{5}+\frac{836083634079}{394631354879}a^{4}+\frac{1003877319184}{394631354879}a^{3}+\frac{496423448226}{394631354879}a^{2}-\frac{1630702493711}{394631354879}a-\frac{182351736196}{394631354879}$, $\frac{21252554341}{394631354879}a^{15}+\frac{13555497035}{394631354879}a^{14}-\frac{356515818642}{394631354879}a^{13}-\frac{355441464091}{394631354879}a^{12}+\frac{1861704330139}{394631354879}a^{11}+\frac{2896817405657}{394631354879}a^{10}-\frac{2905568124092}{394631354879}a^{9}-\frac{8592071560830}{394631354879}a^{8}-\frac{2795450038516}{394631354879}a^{7}+\frac{7495023698492}{394631354879}a^{6}+\frac{7337423255792}{394631354879}a^{5}+\frac{1624313391109}{394631354879}a^{4}+\frac{2120455023626}{394631354879}a^{3}+\frac{1277324708534}{394631354879}a^{2}-\frac{1367729111372}{394631354879}a-\frac{418079885150}{394631354879}$, $\frac{18858516240}{394631354879}a^{15}+\frac{15697117053}{394631354879}a^{14}-\frac{325974413437}{394631354879}a^{13}-\frac{342644630870}{394631354879}a^{12}+\frac{1696814585567}{394631354879}a^{11}+\frac{2688614010137}{394631354879}a^{10}-\frac{2616141098314}{394631354879}a^{9}-\frac{7959007169409}{394631354879}a^{8}-\frac{2562479130839}{394631354879}a^{7}+\frac{7304680879011}{394631354879}a^{6}+\frac{6982735280977}{394631354879}a^{5}+\frac{836083634079}{394631354879}a^{4}+\frac{1003877319184}{394631354879}a^{3}+\frac{496423448226}{394631354879}a^{2}-\frac{1236071138832}{394631354879}a-\frac{182351736196}{394631354879}$, $\frac{153923887948}{394631354879}a^{15}-\frac{231813070193}{394631354879}a^{14}-\frac{1869328102334}{394631354879}a^{13}+\frac{920628086260}{394631354879}a^{12}+\frac{9366934746117}{394631354879}a^{11}+\frac{4005044005440}{394631354879}a^{10}-\frac{19432007545283}{394631354879}a^{9}-\frac{23810675315013}{394631354879}a^{8}+\frac{6622663377854}{394631354879}a^{7}+\frac{28352957105162}{394631354879}a^{6}+\frac{14800580725395}{394631354879}a^{5}+\frac{3276019605989}{394631354879}a^{4}+\frac{3741457566446}{394631354879}a^{3}-\frac{3333271034552}{394631354879}a^{2}-\frac{3403640094424}{394631354879}a+\frac{86238500734}{394631354879}$, $\frac{80981816464}{394631354879}a^{15}-\frac{149724749069}{394631354879}a^{14}-\frac{953280522249}{394631354879}a^{13}+\frac{836909790083}{394631354879}a^{12}+\frac{4887271758447}{394631354879}a^{11}+\frac{410670828140}{394631354879}a^{10}-\frac{11509511481944}{394631354879}a^{9}-\frac{9553804566839}{394631354879}a^{8}+\frac{8660287531806}{394631354879}a^{7}+\frac{15384112824332}{394631354879}a^{6}+\frac{2936859418162}{394631354879}a^{5}-\frac{2040489592775}{394631354879}a^{4}+\frac{715469281699}{394631354879}a^{3}-\frac{2844786981084}{394631354879}a^{2}-\frac{1739502530159}{394631354879}a+\frac{56067320976}{394631354879}$, $\frac{11065574473}{394631354879}a^{15}-\frac{39111046368}{394631354879}a^{14}-\frac{132056670629}{394631354879}a^{13}+\frac{410669342180}{394631354879}a^{12}+\frac{859070314837}{394631354879}a^{11}-\frac{1491880283486}{394631354879}a^{10}-\frac{3512062283889}{394631354879}a^{9}+\frac{1312533512443}{394631354879}a^{8}+\frac{7402425246898}{394631354879}a^{7}+\frac{3581565806373}{394631354879}a^{6}-\frac{5190145658508}{394631354879}a^{5}-\frac{5453642004696}{394631354879}a^{4}-\frac{1554294429722}{394631354879}a^{3}-\frac{1990882413661}{394631354879}a^{2}-\frac{96169935659}{394631354879}a+\frac{1471611148045}{394631354879}$, $\frac{145071726581}{394631354879}a^{15}-\frac{249698861890}{394631354879}a^{14}-\frac{1699990624673}{394631354879}a^{13}+\frac{1221240343804}{394631354879}a^{12}+\frac{8487387719158}{394631354879}a^{11}+\frac{1970244790955}{394631354879}a^{10}-\frac{18477865116888}{394631354879}a^{9}-\frac{18173786769569}{394631354879}a^{8}+\frac{10134864723020}{394631354879}a^{7}+\frac{23817220135672}{394631354879}a^{6}+\frac{7589574428099}{394631354879}a^{5}+\frac{909773606402}{394631354879}a^{4}+\frac{4283402744264}{394631354879}a^{3}-\frac{2347017961111}{394631354879}a^{2}-\frac{2269996774106}{394631354879}a+\frac{302127259120}{394631354879}$, $\frac{123819172240}{394631354879}a^{15}-\frac{263254358925}{394631354879}a^{14}-\frac{1343474806031}{394631354879}a^{13}+\frac{1576681807895}{394631354879}a^{12}+\frac{6625683389019}{394631354879}a^{11}-\frac{926572614702}{394631354879}a^{10}-\frac{15572296992796}{394631354879}a^{9}-\frac{9581715208739}{394631354879}a^{8}+\frac{12930314761536}{394631354879}a^{7}+\frac{16322196437180}{394631354879}a^{6}+\frac{252151172307}{394631354879}a^{5}-\frac{714539784707}{394631354879}a^{4}+\frac{2162947720638}{394631354879}a^{3}-\frac{3624342669645}{394631354879}a^{2}-\frac{507636307855}{394631354879}a+\frac{720207144270}{394631354879}$, $\frac{129994087995}{394631354879}a^{15}-\frac{262382214091}{394631354879}a^{14}-\frac{1425399309826}{394631354879}a^{13}+\frac{1455940715031}{394631354879}a^{12}+\frac{7006471837865}{394631354879}a^{11}+\frac{108307345283}{394631354879}a^{10}-\frac{15785807212208}{394631354879}a^{9}-\frac{12523465120914}{394631354879}a^{8}+\frac{10426653664440}{394631354879}a^{7}+\frac{18405016602137}{394631354879}a^{6}+\frac{4673582329487}{394631354879}a^{5}+\frac{1211445919458}{394631354879}a^{4}+\frac{1900146963362}{394631354879}a^{3}-\frac{3666695107779}{394631354879}a^{2}-\frac{1274016968369}{394631354879}a+\frac{365722236949}{394631354879}$, $\frac{108197574492}{394631354879}a^{15}-\frac{131171762674}{394631354879}a^{14}-\frac{1379900644031}{394631354879}a^{13}+\frac{292714963109}{394631354879}a^{12}+\frac{6979982318171}{394631354879}a^{11}+\frac{4611160040263}{394631354879}a^{10}-\frac{13862647636130}{394631354879}a^{9}-\frac{21086565726498}{394631354879}a^{8}+\frac{2034450221809}{394631354879}a^{7}+\frac{23709094654806}{394631354879}a^{6}+\frac{14839741968907}{394631354879}a^{5}+\frac{2352957111147}{394631354879}a^{4}+\frac{2588155623300}{394631354879}a^{3}-\frac{1639109129856}{394631354879}a^{2}-\frac{3367393311266}{394631354879}a-\frac{198996839330}{394631354879}$ 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:  \( 23241.037751 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{3}\cdot 23241.037751 \cdot 1}{2\cdot\sqrt{298892216010498046875}}\cr\approx \mathstrut & 0.17072918993 \end{aligned}\] (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 - x^15 - 13*x^14 + 65*x^12 + 56*x^11 - 119*x^10 - 220*x^9 - 21*x^8 + 220*x^7 + 179*x^6 + 48*x^5 + 30*x^4 - 5*x^3 - 32*x^2 - 8*x + 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^16 - x^15 - 13*x^14 + 65*x^12 + 56*x^11 - 119*x^10 - 220*x^9 - 21*x^8 + 220*x^7 + 179*x^6 + 48*x^5 + 30*x^4 - 5*x^3 - 32*x^2 - 8*x + 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^16 - x^15 - 13*x^14 + 65*x^12 + 56*x^11 - 119*x^10 - 220*x^9 - 21*x^8 + 220*x^7 + 179*x^6 + 48*x^5 + 30*x^4 - 5*x^3 - 32*x^2 - 8*x + 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^16 - x^15 - 13*x^14 + 65*x^12 + 56*x^11 - 119*x^10 - 220*x^9 - 21*x^8 + 220*x^7 + 179*x^6 + 48*x^5 + 30*x^4 - 5*x^3 - 32*x^2 - 8*x + 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

$C_4^4.C_2\wr C_4$ (as 16T1771):

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 16384
The 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$
Character table for $C_4^4.C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.8.1292203125.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.4.52401279790283203125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ R R $16$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ $16$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(3\) Copy content Toggle raw display 3.8.2.8a1.2$x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(5\) Copy content Toggle raw display 5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(179\) Copy content Toggle raw display $\Q_{179}$$x + 177$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{179}$$x + 177$$1$$1$$0$Trivial$$[\ ]$$
179.2.1.0a1.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
179.2.1.0a1.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
179.1.2.1a1.2$x^{2} + 358$$2$$1$$1$$C_2$$$[\ ]_{2}$$
179.2.1.0a1.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
179.2.1.0a1.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
179.4.1.0a1.1$x^{4} + x^{2} + 109 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(1021\) Copy content Toggle raw display $\Q_{1021}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1021}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
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)