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Properties

Label 20.0.488...000.1
Degree $20$
Signature $(0, 10)$
Discriminant $4.883\times 10^{23}$
Root discriminant \(15.29\)
Ramified primes $2,5$
Class number $1$
Class group trivial
Galois group $C_4\times D_5$ (as 20T6)

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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^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 10*y^19 + 50*y^18 - 155*y^17 + 325*y^16 - 464*y^15 + 420*y^14 - 140*y^13 - 205*y^12 + 350*y^11 - 194*y^10 - 85*y^9 + 220*y^8 - 170*y^7 + 45*y^6 + 11*y^5 - 5*y^3 + 5*y^2 + 5*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*x + 1)
 

\( x^{20} - 10 x^{19} + 50 x^{18} - 155 x^{17} + 325 x^{16} - 464 x^{15} + 420 x^{14} - 140 x^{13} + \cdots + 1 \) 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:  $20$
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:  $(0, 10)$
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:   \(488281250000000000000000\) \(\medspace = 2^{16}\cdot 5^{27}\) 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:  \(15.29\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2\cdot 5^{27/20}\approx 17.564650049460273$
Ramified primes:   \(2\), \(5\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\zeta_{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{48314316844849}a^{19}+\frac{2158374985593}{4392210622259}a^{18}-\frac{19312124365044}{48314316844849}a^{17}-\frac{13485989362534}{48314316844849}a^{16}+\frac{10942425054607}{48314316844849}a^{15}+\frac{920244885600}{48314316844849}a^{14}+\frac{12279019176628}{48314316844849}a^{13}-\frac{15054827396546}{48314316844849}a^{12}-\frac{22244290514076}{48314316844849}a^{11}-\frac{17706819140404}{48314316844849}a^{10}+\frac{17861746782486}{48314316844849}a^{9}+\frac{6680134690978}{48314316844849}a^{8}+\frac{18525576144657}{48314316844849}a^{7}+\frac{14134580911150}{48314316844849}a^{6}-\frac{4628551523965}{48314316844849}a^{5}+\frac{20232909133388}{48314316844849}a^{4}-\frac{9677543067033}{48314316844849}a^{3}+\frac{5178069613806}{48314316844849}a^{2}-\frac{17835699589165}{48314316844849}a+\frac{6894647134673}{48314316844849}$ 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 $1$
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:  Trivial group, which has order $1$
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:  $9$
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:   \( \frac{25647672629082}{48314316844849} a^{19} - \frac{23382991792054}{4392210622259} a^{18} + \frac{1293488166942520}{48314316844849} a^{17} - \frac{4046355370809716}{48314316844849} a^{16} + \frac{8607879987798252}{48314316844849} a^{15} - \frac{12590356237221986}{48314316844849} a^{14} + \frac{11974151847780594}{48314316844849} a^{13} - \frac{4978800801360226}{48314316844849} a^{12} - \frac{4386127065295506}{48314316844849} a^{11} + \frac{9176307093212860}{48314316844849} a^{10} - \frac{5968565866786384}{48314316844849} a^{9} - \frac{1317834458946326}{48314316844849} a^{8} + \frac{5659642664031292}{48314316844849} a^{7} - \frac{5007501121126595}{48314316844849} a^{6} + \frac{1722849221016958}{48314316844849} a^{5} + \frac{106128301747313}{48314316844849} a^{4} - \frac{102704872821264}{48314316844849} a^{3} - \frac{140755295603881}{48314316844849} a^{2} + \frac{185038799661752}{48314316844849} a + \frac{103109586667261}{48314316844849} \)  (order $10$) 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:   $\frac{13446405590412}{48314316844849}a^{19}-\frac{11873604875845}{4392210622259}a^{18}+\frac{636381987339028}{48314316844849}a^{17}-\frac{19\cdots 09}{48314316844849}a^{16}+\frac{39\cdots 88}{48314316844849}a^{15}-\frac{55\cdots 96}{48314316844849}a^{14}+\frac{50\cdots 32}{48314316844849}a^{13}-\frac{22\cdots 75}{48314316844849}a^{12}-\frac{12\cdots 52}{48314316844849}a^{11}+\frac{28\cdots 96}{48314316844849}a^{10}-\frac{18\cdots 24}{48314316844849}a^{9}-\frac{396126322722699}{48314316844849}a^{8}+\frac{16\cdots 06}{48314316844849}a^{7}-\frac{15\cdots 29}{48314316844849}a^{6}+\frac{796149299576888}{48314316844849}a^{5}-\frac{374357899408112}{48314316844849}a^{4}+\frac{307826316743642}{48314316844849}a^{3}-\frac{157751916690794}{48314316844849}a^{2}+\frac{101457233328100}{48314316844849}a+\frac{59873997144261}{48314316844849}$, $\frac{20826152778168}{48314316844849}a^{19}-\frac{19620699480784}{4392210622259}a^{18}+\frac{11\cdots 35}{48314316844849}a^{17}-\frac{36\cdots 92}{48314316844849}a^{16}+\frac{82\cdots 06}{48314316844849}a^{15}-\frac{13\cdots 12}{48314316844849}a^{14}+\frac{14\cdots 21}{48314316844849}a^{13}-\frac{89\cdots 33}{48314316844849}a^{12}-\frac{589662054695850}{48314316844849}a^{11}+\frac{79\cdots 22}{48314316844849}a^{10}-\frac{79\cdots 95}{48314316844849}a^{9}+\frac{18\cdots 94}{48314316844849}a^{8}+\frac{41\cdots 86}{48314316844849}a^{7}-\frac{57\cdots 07}{48314316844849}a^{6}+\frac{35\cdots 70}{48314316844849}a^{5}-\frac{11\cdots 91}{48314316844849}a^{4}+\frac{189114756219845}{48314316844849}a^{3}-\frac{71446422896782}{48314316844849}a^{2}+\frac{141272217352292}{48314316844849}a+\frac{44230381417498}{48314316844849}$, $\frac{7360814356487}{48314316844849}a^{19}-\frac{6900669925641}{4392210622259}a^{18}+\frac{386168676959119}{48314316844849}a^{17}-\frac{12\cdots 80}{48314316844849}a^{16}+\frac{25\cdots 80}{48314316844849}a^{15}-\frac{37\cdots 52}{48314316844849}a^{14}+\frac{36\cdots 61}{48314316844849}a^{13}-\frac{16\cdots 60}{48314316844849}a^{12}-\frac{10\cdots 70}{48314316844849}a^{11}+\frac{26\cdots 43}{48314316844849}a^{10}-\frac{21\cdots 52}{48314316844849}a^{9}+\frac{145372960268809}{48314316844849}a^{8}+\frac{13\cdots 68}{48314316844849}a^{7}-\frac{16\cdots 26}{48314316844849}a^{6}+\frac{927228409587954}{48314316844849}a^{5}-\frac{136626947829208}{48314316844849}a^{4}-\frac{37828302911915}{48314316844849}a^{3}+\frac{24696517847835}{48314316844849}a^{2}+\frac{44112281186482}{48314316844849}a+\frac{40091046913157}{48314316844849}$, $a$, $\frac{9177905156210}{48314316844849}a^{19}-\frac{7747013686013}{4392210622259}a^{18}+\frac{390687225516912}{48314316844849}a^{17}-\frac{10\cdots 01}{48314316844849}a^{16}+\frac{18\cdots 12}{48314316844849}a^{15}-\frac{16\cdots 79}{48314316844849}a^{14}-\frac{154521664556848}{48314316844849}a^{13}+\frac{28\cdots 35}{48314316844849}a^{12}-\frac{41\cdots 08}{48314316844849}a^{11}+\frac{25\cdots 70}{48314316844849}a^{10}+\frac{630468411824997}{48314316844849}a^{9}-\frac{26\cdots 16}{48314316844849}a^{8}+\frac{19\cdots 84}{48314316844849}a^{7}-\frac{195455056770838}{48314316844849}a^{6}-\frac{10\cdots 36}{48314316844849}a^{5}+\frac{853121938931278}{48314316844849}a^{4}-\frac{318950336500669}{48314316844849}a^{3}+\frac{85413381354707}{48314316844849}a^{2}-\frac{46037703019444}{48314316844849}a+\frac{49467908436451}{48314316844849}$, $\frac{3263020430366}{48314316844849}a^{19}-\frac{4045994578309}{4392210622259}a^{18}+\frac{286304928043873}{48314316844849}a^{17}-\frac{11\cdots 85}{48314316844849}a^{16}+\frac{31\cdots 83}{48314316844849}a^{15}-\frac{61\cdots 53}{48314316844849}a^{14}+\frac{85\cdots 51}{48314316844849}a^{13}-\frac{79\cdots 64}{48314316844849}a^{12}+\frac{33\cdots 08}{48314316844849}a^{11}+\frac{26\cdots 34}{48314316844849}a^{10}-\frac{58\cdots 60}{48314316844849}a^{9}+\frac{40\cdots 02}{48314316844849}a^{8}+\frac{393630604677350}{48314316844849}a^{7}-\frac{34\cdots 01}{48314316844849}a^{6}+\frac{34\cdots 26}{48314316844849}a^{5}-\frac{16\cdots 27}{48314316844849}a^{4}+\frac{272244564416988}{48314316844849}a^{3}+\frac{143717356613824}{48314316844849}a^{2}-\frac{4046010491639}{48314316844849}a-\frac{34758919600258}{48314316844849}$, $\frac{2520761246760}{48314316844849}a^{19}-\frac{1703985075757}{4392210622259}a^{18}+\frac{62062992167296}{48314316844849}a^{17}-\frac{75957996034588}{48314316844849}a^{16}-\frac{133533519240037}{48314316844849}a^{15}+\frac{759349101269734}{48314316844849}a^{14}-\frac{15\cdots 75}{48314316844849}a^{13}+\frac{17\cdots 64}{48314316844849}a^{12}-\frac{961845319867743}{48314316844849}a^{11}-\frac{409486932474943}{48314316844849}a^{10}+\frac{11\cdots 67}{48314316844849}a^{9}-\frac{767089023706227}{48314316844849}a^{8}-\frac{259573711045658}{48314316844849}a^{7}+\frac{730494192373206}{48314316844849}a^{6}-\frac{562710321207693}{48314316844849}a^{5}+\frac{210271141949593}{48314316844849}a^{4}-\frac{64674716761060}{48314316844849}a^{3}+\frac{126387580080276}{48314316844849}a^{2}-\frac{55188447784082}{48314316844849}a-\frac{16024971839802}{48314316844849}$, $\frac{17673836969969}{48314316844849}a^{19}-\frac{17077035430165}{4392210622259}a^{18}+\frac{990560094686032}{48314316844849}a^{17}-\frac{32\cdots 67}{48314316844849}a^{16}+\frac{72\cdots 80}{48314316844849}a^{15}-\frac{11\cdots 35}{48314316844849}a^{14}+\frac{11\cdots 70}{48314316844849}a^{13}-\frac{62\cdots 30}{48314316844849}a^{12}-\frac{24\cdots 34}{48314316844849}a^{11}+\frac{80\cdots 79}{48314316844849}a^{10}-\frac{65\cdots 37}{48314316844849}a^{9}+\frac{284586345762580}{48314316844849}a^{8}+\frac{45\cdots 65}{48314316844849}a^{7}-\frac{49\cdots 64}{48314316844849}a^{6}+\frac{25\cdots 00}{48314316844849}a^{5}-\frac{360538836445082}{48314316844849}a^{4}-\frac{119296487929445}{48314316844849}a^{3}+\frac{83750893732444}{48314316844849}a^{2}+\frac{81267024088860}{48314316844849}a+\frac{12660463475829}{48314316844849}$, $\frac{9238269534386}{48314316844849}a^{19}-\frac{9987562726696}{4392210622259}a^{18}+\frac{623517806040130}{48314316844849}a^{17}-\frac{21\cdots 61}{48314316844849}a^{16}+\frac{50\cdots 48}{48314316844849}a^{15}-\frac{81\cdots 70}{48314316844849}a^{14}+\frac{83\cdots 94}{48314316844849}a^{13}-\frac{39\cdots 94}{48314316844849}a^{12}-\frac{28\cdots 47}{48314316844849}a^{11}+\frac{66\cdots 36}{48314316844849}a^{10}-\frac{44\cdots 11}{48314316844849}a^{9}-\frac{11\cdots 54}{48314316844849}a^{8}+\frac{44\cdots 39}{48314316844849}a^{7}-\frac{33\cdots 74}{48314316844849}a^{6}+\frac{859910847031367}{48314316844849}a^{5}+\frac{497742879490517}{48314316844849}a^{4}-\frac{76388047168775}{48314316844849}a^{3}-\frac{365365477706854}{48314316844849}a^{2}+\frac{104193825560129}{48314316844849}a+\frac{62764790477501}{48314316844849}$ Copy content Toggle raw display
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:  \( 22883.422779 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 22883.422779 \cdot 1}{10\cdot\sqrt{488281250000000000000000}}\cr\approx \mathstrut & 0.31403976532 \end{aligned}\]

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^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*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^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*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^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*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 = polynomial_ring(QQ); K, a = number_field(x^20 - 10*x^19 + 50*x^18 - 155*x^17 + 325*x^16 - 464*x^15 + 420*x^14 - 140*x^13 - 205*x^12 + 350*x^11 - 194*x^10 - 85*x^9 + 220*x^8 - 170*x^7 + 45*x^6 + 11*x^5 - 5*x^3 + 5*x^2 + 5*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

$C_4\times D_5$ (as 20T6):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_4\times D_5$
Character table for $C_4\times D_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.250000.1, 10.2.312500000000.1

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(L)]
 
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 20 sibling: 20.4.7812500000000000000000000.1
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 $20$ R $20$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{5}$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ $20$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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 2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.4.2.8a1.1$x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$$2$$4$$8$$C_4\times C_2$$$[2]^{4}$$
2.4.2.8a1.1$x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$$2$$4$$8$$C_4\times C_2$$$[2]^{4}$$
\(5\) Copy content Toggle raw display 5.1.20.27a1.1$x^{20} + 15 x^{8} + 5$$20$$1$$27$20T2not computed

Spectrum of ring of integers

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