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Properties

Label 20.0.488...125.1
Degree $20$
Signature $(0, 10)$
Discriminant $4.888\times 10^{22}$
Root discriminant \(13.63\)
Ramified primes $3,5$
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 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*x^2 + 5*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 10*y^19 + 40*y^18 - 75*y^17 + 45*y^16 + 48*y^15 - 5*y^14 - 295*y^13 + 535*y^12 - 285*y^11 - 261*y^10 + 375*y^9 + 50*y^8 - 275*y^7 + 65*y^6 + 92*y^5 - 35*y^4 - 25*y^3 + 10*y^2 + 5*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*x^2 + 5*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 10*x^19 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*x^2 + 5*x + 1)
 

\( x^{20} - 10 x^{19} + 40 x^{18} - 75 x^{17} + 45 x^{16} + 48 x^{15} - 5 x^{14} - 295 x^{13} + 535 x^{12} + \cdots + 1 \) 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:   \(48883259296417236328125\) \(\medspace = 3^{8}\cdot 5^{27}\) 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:  \(13.63\)
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:  $3^{1/2}5^{27/20}\approx 15.211433151416195$
Ramified primes:   \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{15}-\frac{1}{9}a^{13}-\frac{1}{9}a^{12}-\frac{1}{9}a^{10}-\frac{4}{9}a^{9}-\frac{1}{3}a^{8}-\frac{2}{9}a^{7}-\frac{2}{9}a^{6}-\frac{2}{9}a^{4}-\frac{2}{9}a^{3}+\frac{1}{3}a^{2}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{1}{27}a^{15}-\frac{4}{27}a^{14}+\frac{1}{27}a^{13}+\frac{2}{27}a^{12}-\frac{1}{27}a^{11}+\frac{1}{27}a^{10}-\frac{10}{27}a^{9}-\frac{5}{27}a^{8}-\frac{10}{27}a^{7}-\frac{2}{27}a^{6}+\frac{1}{27}a^{5}+\frac{5}{27}a^{4}-\frac{2}{27}a^{3}+\frac{1}{27}a^{2}-\frac{13}{27}a-\frac{2}{27}$, $\frac{1}{27}a^{18}-\frac{1}{9}a^{15}-\frac{1}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{27}a^{12}-\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{8}{27}a^{6}-\frac{4}{9}a^{5}+\frac{1}{9}a^{4}-\frac{1}{27}a^{3}-\frac{4}{9}a^{2}+\frac{1}{9}a+\frac{7}{27}$, $\frac{1}{7209}a^{19}+\frac{124}{7209}a^{18}+\frac{34}{2403}a^{17}-\frac{8}{2403}a^{16}+\frac{11}{2403}a^{15}-\frac{379}{2403}a^{14}-\frac{173}{7209}a^{13}+\frac{553}{7209}a^{12}+\frac{137}{2403}a^{11}+\frac{374}{2403}a^{10}-\frac{701}{2403}a^{9}-\frac{626}{2403}a^{8}-\frac{88}{7209}a^{7}-\frac{52}{7209}a^{6}+\frac{814}{2403}a^{5}+\frac{2915}{7209}a^{4}+\frac{2891}{7209}a^{3}-\frac{728}{2403}a^{2}+\frac{520}{7209}a+\frac{1600}{7209}$ 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:   \( \frac{5327}{7209} a^{19} - \frac{55546}{7209} a^{18} + \frac{2947}{89} a^{17} - \frac{172022}{2403} a^{16} + \frac{167444}{2403} a^{15} - \frac{214}{267} a^{14} - \frac{35932}{7209} a^{13} - \frac{1505059}{7209} a^{12} + \frac{44022}{89} a^{11} - \frac{1111933}{2403} a^{10} + \frac{111463}{2403} a^{9} + \frac{209786}{801} a^{8} - \frac{861800}{7209} a^{7} - \frac{877220}{7209} a^{6} + \frac{84553}{801} a^{5} + \frac{76381}{7209} a^{4} - \frac{208997}{7209} a^{3} - \frac{2482}{801} a^{2} + \frac{44504}{7209} a + \frac{7502}{7209} \)  (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:   $\frac{3308}{7209}a^{19}-\frac{31426}{7209}a^{18}+\frac{41272}{2403}a^{17}-\frac{9299}{267}a^{16}+\frac{9482}{267}a^{15}-\frac{34699}{2403}a^{14}+\frac{158495}{7209}a^{13}-\frac{792883}{7209}a^{12}+\frac{546023}{2403}a^{11}-\frac{191972}{801}a^{10}+\frac{91814}{801}a^{9}+\frac{22294}{2403}a^{8}-\frac{185639}{7209}a^{7}-\frac{80168}{7209}a^{6}+\frac{56176}{2403}a^{5}-\frac{84791}{7209}a^{4}+\frac{31261}{7209}a^{3}-\frac{7271}{2403}a^{2}+\frac{6554}{7209}a+\frac{3797}{7209}$, $\frac{3317}{7209}a^{19}-\frac{32980}{7209}a^{18}+\frac{44515}{2403}a^{17}-\frac{29434}{801}a^{16}+\frac{7802}{267}a^{15}+\frac{26504}{2403}a^{14}-\frac{53725}{7209}a^{13}-\frac{816208}{7209}a^{12}+\frac{605195}{2403}a^{11}-\frac{159166}{801}a^{10}-\frac{38449}{801}a^{9}+\frac{485512}{2403}a^{8}-\frac{725504}{7209}a^{7}-\frac{381545}{7209}a^{6}+\frac{145204}{2403}a^{5}-\frac{17705}{7209}a^{4}-\frac{101852}{7209}a^{3}+\frac{17416}{2403}a^{2}-\frac{5587}{7209}a-\frac{1294}{7209}$, $\frac{3311}{7209}a^{19}-\frac{28384}{7209}a^{18}+\frac{30338}{2403}a^{17}-\frac{36634}{2403}a^{16}-\frac{13241}{2403}a^{15}+\frac{52897}{2403}a^{14}+\frac{173729}{7209}a^{13}-\frac{744232}{7209}a^{12}+\frac{227458}{2403}a^{11}+\frac{82204}{2403}a^{10}-\frac{292078}{2403}a^{9}+\frac{76664}{2403}a^{8}+\frac{545677}{7209}a^{7}-\frac{283511}{7209}a^{6}-\frac{52009}{2403}a^{5}+\frac{102844}{7209}a^{4}+\frac{45808}{7209}a^{3}-\frac{6340}{2403}a^{2}-\frac{24460}{7209}a-\frac{5020}{7209}$, $\frac{734}{7209}a^{19}-\frac{15250}{7209}a^{18}+\frac{33233}{2403}a^{17}-\frac{100390}{2403}a^{16}+\frac{140506}{2403}a^{15}-\frac{41624}{2403}a^{14}-\frac{210286}{7209}a^{13}-\frac{482941}{7209}a^{12}+\frac{709051}{2403}a^{11}-\frac{918974}{2403}a^{10}+\frac{323846}{2403}a^{9}+\frac{438170}{2403}a^{8}-\frac{1214027}{7209}a^{7}-\frac{321989}{7209}a^{6}+\frac{235424}{2403}a^{5}-\frac{135230}{7209}a^{4}-\frac{173138}{7209}a^{3}+\frac{20741}{2403}a^{2}+\frac{38852}{7209}a-\frac{400}{7209}$, $\frac{287}{801}a^{19}-\frac{2237}{801}a^{18}+\frac{18491}{2403}a^{17}-\frac{14879}{2403}a^{16}-\frac{17956}{2403}a^{15}+\frac{16063}{2403}a^{14}+\frac{72746}{2403}a^{13}-\frac{139391}{2403}a^{12}+\frac{55543}{2403}a^{11}+\frac{95177}{2403}a^{10}-\frac{73403}{2403}a^{9}-\frac{101644}{2403}a^{8}+\frac{81406}{2403}a^{7}+\frac{82943}{2403}a^{6}-\frac{62986}{2403}a^{5}-\frac{31046}{2403}a^{4}+\frac{31238}{2403}a^{3}+\frac{7889}{2403}a^{2}-\frac{13745}{2403}a-\frac{3235}{2403}$, $\frac{3673}{7209}a^{19}-\frac{31022}{7209}a^{18}+\frac{32500}{2403}a^{17}-\frac{37661}{2403}a^{16}-\frac{16201}{2403}a^{15}+\frac{52670}{2403}a^{14}+\frac{212830}{7209}a^{13}-\frac{783812}{7209}a^{12}+\frac{219113}{2403}a^{11}+\frac{101981}{2403}a^{10}-\frac{285782}{2403}a^{9}+\frac{43894}{2403}a^{8}+\frac{525035}{7209}a^{7}-\frac{93808}{7209}a^{6}-\frac{94829}{2403}a^{5}+\frac{98351}{7209}a^{4}+\frac{89765}{7209}a^{3}-\frac{11420}{2403}a^{2}-\frac{38072}{7209}a-\frac{1205}{7209}$, $\frac{3602}{7209}a^{19}-\frac{27277}{7209}a^{18}+\frac{21631}{2403}a^{17}+\frac{289}{267}a^{16}-\frac{28949}{801}a^{15}+\frac{94442}{2403}a^{14}+\frac{287858}{7209}a^{13}-\frac{666613}{7209}a^{12}-\frac{71320}{2403}a^{11}+\frac{188665}{801}a^{10}-\frac{185945}{801}a^{9}-\frac{57887}{2403}a^{8}+\frac{1030840}{7209}a^{7}+\frac{29767}{7209}a^{6}-\frac{192050}{2403}a^{5}+\frac{109792}{7209}a^{4}+\frac{153373}{7209}a^{3}-\frac{7169}{2403}a^{2}-\frac{72856}{7209}a-\frac{12277}{7209}$, $\frac{7834}{7209}a^{19}-\frac{67748}{7209}a^{18}+\frac{24913}{801}a^{17}-\frac{103612}{2403}a^{16}+\frac{16309}{2403}a^{15}+\frac{20158}{801}a^{14}+\frac{482479}{7209}a^{13}-\frac{1712678}{7209}a^{12}+\frac{208826}{801}a^{11}-\frac{126611}{2403}a^{10}-\frac{272140}{2403}a^{9}-\frac{198}{89}a^{8}+\frac{899525}{7209}a^{7}-\frac{182554}{7209}a^{6}-\frac{43720}{801}a^{5}+\frac{68219}{7209}a^{4}+\frac{181379}{7209}a^{3}-\frac{5855}{801}a^{2}-\frac{44528}{7209}a-\frac{9260}{7209}$, $\frac{5155}{7209}a^{19}-\frac{44300}{7209}a^{18}+\frac{16267}{801}a^{17}-\frac{67228}{2403}a^{16}+\frac{3928}{2403}a^{15}+\frac{22630}{801}a^{14}+\frac{182593}{7209}a^{13}-\frac{1097147}{7209}a^{12}+\frac{148607}{801}a^{11}-\frac{44537}{2403}a^{10}-\frac{390520}{2403}a^{9}+\frac{29491}{267}a^{8}+\frac{401561}{7209}a^{7}-\frac{580717}{7209}a^{6}-\frac{772}{801}a^{5}+\frac{236093}{7209}a^{4}-\frac{46225}{7209}a^{3}-\frac{4472}{801}a^{2}+\frac{3391}{7209}a+\frac{1705}{7209}$ 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:  \( 6025.48578502 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 6025.48578502 \cdot 1}{10\cdot\sqrt{48883259296417236328125}}\cr\approx \mathstrut & 0.261342879804 \end{aligned}\]

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 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*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 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*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 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*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 + 40*x^18 - 75*x^17 + 45*x^16 + 48*x^15 - 5*x^14 - 295*x^13 + 535*x^12 - 285*x^11 - 261*x^10 + 375*x^9 + 50*x^8 - 275*x^7 + 65*x^6 + 92*x^5 - 35*x^4 - 25*x^3 + 10*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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.140625.1, 10.2.98876953125.1

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: 20.4.439949333667755126953125.1
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 $20$ R R ${\href{/padicField/7.4.0.1}{4} }^{5}$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/29.2.0.1}{2} }^{10}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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
\(3\) Copy content Toggle raw display 3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.1.20.27a4.1$x^{20} + 10 x^{8} + 5$$20$$1$$27$not computednot 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)