↑

Properties

Label 20.0.623...125.1
Degree $20$
Signature $(0, 10)$
Discriminant $6.239\times 10^{32}$
Root discriminant \(43.63\)
Ramified primes $5,11$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $C_4\times D_5$ (as 20T6)

Related objects

Downloads

Learn more

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 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880)
 
Copy content gp:K = bnfinit(y^20 - 20*y^18 + 170*y^16 - 800*y^14 + 2275*y^12 - 3480*y^10 - 950*y^8 + 15700*y^6 - 25375*y^4 + 13000*y^2 + 28880, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880)
 

\( x^{20} - 20 x^{18} + 170 x^{16} - 800 x^{14} + 2275 x^{12} - 3480 x^{10} - 950 x^{8} + 15700 x^{6} + \cdots + 28880 \) 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:   \(623866452951915562152862548828125\) \(\medspace = 5^{35}\cdot 11^{8}\) 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:  \(43.63\)
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:  $5^{7/4}11^{1/2}\approx 55.449016844865795$
Ramified primes:   \(5\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{5}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{176}a^{10}-\frac{5}{88}a^{8}-\frac{9}{176}a^{6}-\frac{3}{88}a^{4}-\frac{19}{176}a^{2}-\frac{1}{2}a-\frac{1}{44}$, $\frac{1}{352}a^{11}-\frac{5}{176}a^{9}+\frac{35}{352}a^{7}-\frac{1}{8}a^{6}+\frac{19}{176}a^{5}-\frac{1}{8}a^{4}-\frac{63}{352}a^{3}+\frac{3}{8}a^{2}-\frac{23}{88}a$, $\frac{1}{352}a^{12}-\frac{21}{352}a^{8}+\frac{9}{88}a^{6}-\frac{35}{352}a^{4}+\frac{57}{176}a^{2}-\frac{1}{2}a+\frac{17}{44}$, $\frac{1}{704}a^{13}-\frac{1}{704}a^{12}-\frac{1}{704}a^{11}+\frac{3}{64}a^{9}+\frac{21}{704}a^{8}-\frac{43}{704}a^{7}-\frac{9}{176}a^{6}-\frac{29}{704}a^{5}-\frac{141}{704}a^{4}-\frac{175}{704}a^{3}-\frac{57}{352}a^{2}-\frac{9}{176}a-\frac{17}{88}$, $\frac{1}{704}a^{14}-\frac{1}{704}a^{11}+\frac{1}{704}a^{10}-\frac{17}{352}a^{9}-\frac{1}{88}a^{8}-\frac{79}{704}a^{7}+\frac{31}{704}a^{6}+\frac{3}{352}a^{5}-\frac{53}{352}a^{4}-\frac{25}{704}a^{3}+\frac{167}{352}a^{2}-\frac{87}{176}a-\frac{1}{8}$, $\frac{1}{13376}a^{15}+\frac{1}{3344}a^{13}-\frac{1}{704}a^{12}-\frac{5}{13376}a^{11}-\frac{1}{352}a^{10}-\frac{35}{836}a^{9}+\frac{41}{704}a^{8}-\frac{1355}{13376}a^{7}-\frac{9}{352}a^{6}-\frac{545}{6688}a^{5}+\frac{47}{704}a^{4}-\frac{41}{1672}a^{3}+\frac{69}{176}a^{2}+\frac{305}{836}a+\frac{7}{22}$, $\frac{1}{26752}a^{16}-\frac{15}{26752}a^{14}+\frac{7}{13376}a^{12}-\frac{1}{704}a^{11}+\frac{29}{26752}a^{10}-\frac{17}{352}a^{9}-\frac{497}{13376}a^{8}+\frac{9}{704}a^{7}-\frac{3123}{26752}a^{6}-\frac{41}{352}a^{5}+\frac{4061}{26752}a^{4}-\frac{113}{704}a^{3}-\frac{205}{6688}a^{2}+\frac{45}{176}a-\frac{2}{11}$, $\frac{1}{26752}a^{17}-\frac{1}{26752}a^{15}-\frac{3}{13376}a^{13}+\frac{35}{26752}a^{11}-\frac{655}{13376}a^{9}-\frac{2105}{26752}a^{7}-\frac{1}{8}a^{6}+\frac{1037}{26752}a^{5}-\frac{1}{8}a^{4}-\frac{343}{1672}a^{3}+\frac{3}{8}a^{2}-\frac{461}{1672}a-\frac{1}{2}$, $\frac{1}{53504}a^{18}-\frac{1}{53504}a^{17}+\frac{1}{53504}a^{15}+\frac{17}{53504}a^{14}+\frac{3}{26752}a^{13}-\frac{27}{53504}a^{12}-\frac{35}{53504}a^{11}-\frac{27}{53504}a^{10}-\frac{1017}{26752}a^{9}-\frac{591}{53504}a^{8}+\frac{5449}{53504}a^{7}-\frac{303}{13376}a^{6}+\frac{2307}{53504}a^{5}+\frac{3285}{53504}a^{4}+\frac{761}{3344}a^{3}-\frac{65}{6688}a^{2}-\frac{375}{3344}a+\frac{5}{16}$, $\frac{1}{53504}a^{19}-\frac{1}{53504}a^{17}-\frac{1}{53504}a^{16}-\frac{1}{26752}a^{15}-\frac{23}{53504}a^{14}-\frac{25}{53504}a^{13}+\frac{31}{26752}a^{12}-\frac{1}{1408}a^{11}+\frac{85}{53504}a^{10}-\frac{2293}{53504}a^{9}+\frac{763}{26752}a^{8}+\frac{1317}{53504}a^{7}+\frac{3313}{53504}a^{6}-\frac{391}{13376}a^{5}-\frac{3605}{53504}a^{4}-\frac{2115}{13376}a^{3}+\frac{541}{1672}a^{2}+\frac{1087}{3344}a+\frac{59}{176}$ 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:  $2$, $11$

Class group and class number

Ideal class group:  $C_{5}$, which has order $5$ (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:  $C_{5}$, which has order $5$ (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:  $9$
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:   \( \frac{5}{13376} a^{15} - \frac{75}{13376} a^{13} + \frac{225}{6688} a^{11} - \frac{1}{352} a^{10} - \frac{125}{1216} a^{9} + \frac{5}{176} a^{8} + \frac{1125}{6688} a^{7} - \frac{35}{352} a^{6} + \frac{41}{13376} a^{5} + \frac{25}{176} a^{4} - \frac{8955}{13376} a^{3} - \frac{25}{352} a^{2} + \frac{2395}{3344} a - \frac{43}{88} \)  (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:   $\frac{1}{13376}a^{15}-\frac{15}{13376}a^{13}+\frac{45}{6688}a^{11}-\frac{1}{352}a^{10}-\frac{25}{1216}a^{9}+\frac{5}{176}a^{8}+\frac{225}{6688}a^{7}-\frac{35}{352}a^{6}+\frac{677}{13376}a^{5}+\frac{25}{176}a^{4}-\frac{5135}{13376}a^{3}-\frac{25}{352}a^{2}+\frac{1315}{3344}a+\frac{1}{88}$, $\frac{3}{26752}a^{19}+\frac{17}{26752}a^{18}-\frac{5}{3344}a^{17}-\frac{127}{13376}a^{16}+\frac{211}{26752}a^{15}+\frac{1553}{26752}a^{14}-\frac{577}{26752}a^{13}-\frac{5079}{26752}a^{12}+\frac{921}{26752}a^{11}+\frac{9693}{26752}a^{10}+\frac{679}{26752}a^{9}-\frac{2443}{26752}a^{8}-\frac{2847}{13376}a^{7}-\frac{575}{418}a^{6}+\frac{5915}{26752}a^{5}+\frac{63803}{26752}a^{4}-\frac{141}{704}a^{3}-\frac{13315}{6688}a^{2}-\frac{1963}{3344}a-\frac{71}{22}$, $\frac{1}{4864}a^{19}-\frac{1}{2432}a^{18}-\frac{17}{4864}a^{17}+\frac{19}{2816}a^{16}+\frac{639}{26752}a^{15}-\frac{2407}{53504}a^{14}-\frac{4603}{53504}a^{13}+\frac{97}{608}a^{12}+\frac{251}{1408}a^{11}-\frac{17475}{53504}a^{10}-\frac{541}{4864}a^{9}+\frac{587}{3344}a^{8}-\frac{30733}{53504}a^{7}+\frac{57463}{53504}a^{6}+\frac{15307}{13376}a^{5}-\frac{111581}{53504}a^{4}-\frac{18019}{13376}a^{3}+\frac{490}{209}a^{2}-\frac{6293}{3344}a+\frac{523}{176}$, $\frac{3}{53504}a^{19}+\frac{1}{13376}a^{18}-\frac{67}{53504}a^{17}-\frac{89}{53504}a^{16}+\frac{299}{26752}a^{15}+\frac{833}{53504}a^{14}-\frac{2623}{53504}a^{13}-\frac{2007}{26752}a^{12}+\frac{135}{1408}a^{11}+\frac{9509}{53504}a^{10}+\frac{357}{53504}a^{9}-\frac{2283}{26752}a^{8}-\frac{18621}{53504}a^{7}-\frac{28811}{53504}a^{6}+\frac{1621}{3344}a^{5}+\frac{3669}{2816}a^{4}-\frac{131}{704}a^{3}-\frac{1245}{3344}a^{2}-\frac{1351}{3344}a-\frac{23}{16}$, $\frac{3}{53504}a^{19}-\frac{1}{13376}a^{18}-\frac{67}{53504}a^{17}+\frac{89}{53504}a^{16}+\frac{299}{26752}a^{15}-\frac{833}{53504}a^{14}-\frac{2623}{53504}a^{13}+\frac{2007}{26752}a^{12}+\frac{135}{1408}a^{11}-\frac{9509}{53504}a^{10}+\frac{357}{53504}a^{9}+\frac{2283}{26752}a^{8}-\frac{18621}{53504}a^{7}+\frac{28811}{53504}a^{6}+\frac{1621}{3344}a^{5}-\frac{3669}{2816}a^{4}-\frac{131}{704}a^{3}+\frac{1245}{3344}a^{2}-\frac{1351}{3344}a+\frac{23}{16}$, $\frac{13}{53504}a^{19}-\frac{3}{53504}a^{18}-\frac{65}{13376}a^{17}+\frac{49}{53504}a^{16}+\frac{2217}{53504}a^{15}-\frac{73}{13376}a^{14}-\frac{10543}{53504}a^{13}+\frac{387}{53504}a^{12}+\frac{31421}{53504}a^{11}+\frac{1031}{13376}a^{10}-\frac{59971}{53504}a^{9}-\frac{2491}{4864}a^{8}+\frac{3529}{3344}a^{7}+\frac{87235}{53504}a^{6}+\frac{16901}{53504}a^{5}-\frac{7135}{2432}a^{4}-\frac{3557}{6688}a^{3}+\frac{2873}{3344}a^{2}+\frac{2561}{3344}a+\frac{323}{88}$, $\frac{1}{4864}a^{19}+\frac{17}{53504}a^{18}-\frac{27}{6688}a^{17}-\frac{371}{53504}a^{16}+\frac{1675}{53504}a^{15}+\frac{35}{608}a^{14}-\frac{6909}{53504}a^{13}-\frac{13405}{53504}a^{12}+\frac{17095}{53504}a^{11}+\frac{4369}{6688}a^{10}-\frac{21281}{53504}a^{9}-\frac{49393}{53504}a^{8}-\frac{579}{1216}a^{7}-\frac{35373}{53504}a^{6}+\frac{97131}{53504}a^{5}+\frac{95977}{26752}a^{4}-\frac{10229}{3344}a^{3}-\frac{22175}{3344}a^{2}-\frac{479}{3344}a+\frac{109}{88}$, $\frac{7}{53504}a^{19}+\frac{3}{26752}a^{18}-\frac{129}{53504}a^{17}-\frac{85}{53504}a^{16}+\frac{259}{13376}a^{15}+\frac{427}{53504}a^{14}-\frac{5003}{53504}a^{13}-\frac{205}{13376}a^{12}+\frac{2197}{6688}a^{11}+\frac{527}{53504}a^{10}-\frac{47439}{53504}a^{9}-\frac{113}{13376}a^{8}+\frac{8423}{4864}a^{7}+\frac{3621}{53504}a^{6}-\frac{64455}{26752}a^{5}-\frac{47315}{53504}a^{4}+\frac{19559}{13376}a^{3}+\frac{137}{76}a^{2}+\frac{8919}{3344}a+\frac{301}{176}$, $\frac{5}{53504}a^{19}+\frac{1}{1408}a^{18}-\frac{113}{53504}a^{17}-\frac{751}{53504}a^{16}+\frac{543}{26752}a^{15}+\frac{6401}{53504}a^{14}-\frac{5785}{53504}a^{13}-\frac{971}{1672}a^{12}+\frac{9377}{26752}a^{11}+\frac{95413}{53504}a^{10}-\frac{36349}{53504}a^{9}-\frac{5501}{1672}a^{8}+\frac{22417}{53504}a^{7}+\frac{84411}{53504}a^{6}+\frac{21631}{13376}a^{5}+\frac{436043}{53504}a^{4}-\frac{56237}{13376}a^{3}-\frac{69797}{3344}a^{2}+\frac{16653}{3344}a+\frac{349}{16}$ 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:  \( 403949827.0826757 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 403949827.0826757 \cdot 5}{10\cdot\sqrt{623866452951915562152862548828125}}\cr\approx \mathstrut & 0.775443746442569 \end{aligned}\] (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^20 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880) 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 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880, 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 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880); 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 - 20*x^18 + 170*x^16 - 800*x^14 + 2275*x^12 - 3480*x^10 - 950*x^8 + 15700*x^6 - 25375*x^4 + 13000*x^2 + 28880); 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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.47265625.1, 10.2.11170196533203125.1

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: deg 40
Degree 20 sibling: 20.4.75487840807181783020496368408203125.1
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 ${\href{/padicField/2.4.0.1}{4} }^{5}$ $20$ R ${\href{/padicField/7.4.0.1}{4} }^{5}$ R ${\href{/padicField/13.4.0.1}{4} }^{5}$ ${\href{/padicField/17.4.0.1}{4} }^{5}$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ $20$ ${\href{/padicField/29.2.0.1}{2} }^{10}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{5}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }^{5}$ $20$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(5\) Copy content Toggle raw display 5.1.20.35a1.1$x^{20} + 20 x^{16} + 5$$20$$1$$35$not computednot computed
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
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$$$[\ ]_{2}$$
11.1.2.1a1.1$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$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*40 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*40 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.55.2t1.a.a$1$ $ 5 \cdot 11 $ \(\Q(\sqrt{-55}) \) $C_2$ (as 2T1) $1$ $-1$
1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
*40 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.55.4t1.a.b$1$ $ 5 \cdot 11 $ \(\Q(\sqrt{110 -22 \sqrt{5}})\) $C_4$ (as 4T1) $0$ $1$
1.55.4t1.a.a$1$ $ 5 \cdot 11 $ \(\Q(\sqrt{110 -22 \sqrt{5}})\) $C_4$ (as 4T1) $0$ $1$
*40 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
*40 2.6875.10t3.a.b$2$ $ 5^{4} \cdot 11 $ 10.0.122872161865234375.5 $D_{10}$ (as 10T3) $1$ $0$
*40 2.6875.5t2.a.a$2$ $ 5^{4} \cdot 11 $ 5.1.47265625.1 $D_{5}$ (as 5T2) $1$ $0$
*40 2.6875.10t3.a.a$2$ $ 5^{4} \cdot 11 $ 10.0.122872161865234375.5 $D_{10}$ (as 10T3) $1$ $0$
*40 2.6875.5t2.a.b$2$ $ 5^{4} \cdot 11 $ 5.1.47265625.1 $D_{5}$ (as 5T2) $1$ $0$
*40 2.6875.20t6.a.b$2$ $ 5^{4} \cdot 11 $ 20.0.623866452951915562152862548828125.1 $C_4\times D_5$ (as 20T6) $0$ $0$
*40 2.6875.20t6.a.d$2$ $ 5^{4} \cdot 11 $ 20.0.623866452951915562152862548828125.1 $C_4\times D_5$ (as 20T6) $0$ $0$
*40 2.6875.20t6.a.a$2$ $ 5^{4} \cdot 11 $ 20.0.623866452951915562152862548828125.1 $C_4\times D_5$ (as 20T6) $0$ $0$
*40 2.6875.20t6.a.c$2$ $ 5^{4} \cdot 11 $ 20.0.623866452951915562152862548828125.1 $C_4\times D_5$ (as 20T6) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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