Properties

Label 18.10.285...984.2
Degree $18$
Signature $(10, 4)$
Discriminant $2.856\times 10^{27}$
Root discriminant \(33.52\)
Ramified primes $2,37,101$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6:S_3^2$ (as 18T373)

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^18 - 10*x^16 - 21*x^14 + 184*x^12 - x^10 - 778*x^8 + 693*x^6 + 140*x^4 - 176*x^2 - 16)
 
Copy content gp:K = bnfinit(y^18 - 10*y^16 - 21*y^14 + 184*y^12 - y^10 - 778*y^8 + 693*y^6 + 140*y^4 - 176*y^2 - 16, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 10*x^16 - 21*x^14 + 184*x^12 - x^10 - 778*x^8 + 693*x^6 + 140*x^4 - 176*x^2 - 16);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 10*x^16 - 21*x^14 + 184*x^12 - x^10 - 778*x^8 + 693*x^6 + 140*x^4 - 176*x^2 - 16)
 

\( x^{18} - 10x^{16} - 21x^{14} + 184x^{12} - x^{10} - 778x^{8} + 693x^{6} + 140x^{4} - 176x^{2} - 16 \) 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:  $18$
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:  $(10, 4)$
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:   \(2855870434202241384378793984\) \(\medspace = 2^{20}\cdot 37^{6}\cdot 101^{6}\) 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:  \(33.52\)
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:  $2^{19/12}37^{1/2}101^{1/2}\approx 183.1860403967972$
Ramified primes:   \(2\), \(37\), \(101\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{8}+\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{13}-\frac{1}{8}a^{7}+\frac{3}{16}a^{5}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{16}a^{14}-\frac{1}{8}a^{8}+\frac{3}{16}a^{6}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{15}-\frac{1}{4}a^{8}+\frac{1}{16}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{18117584}a^{16}+\frac{185885}{9058792}a^{14}+\frac{499645}{9058792}a^{12}-\frac{438211}{9058792}a^{10}-\frac{481713}{18117584}a^{8}-\frac{150847}{2264698}a^{6}-\frac{4157823}{9058792}a^{4}-\frac{1}{2}a^{3}-\frac{225823}{2264698}a^{2}-\frac{1}{2}a+\frac{204647}{1132349}$, $\frac{1}{36235168}a^{17}-\frac{1}{36235168}a^{16}-\frac{760579}{36235168}a^{15}+\frac{760579}{36235168}a^{14}+\frac{499645}{18117584}a^{13}-\frac{499645}{18117584}a^{12}-\frac{438211}{18117584}a^{11}+\frac{438211}{18117584}a^{10}+\frac{1782985}{36235168}a^{9}+\frac{7275807}{36235168}a^{8}+\frac{4454969}{36235168}a^{7}+\frac{4603823}{36235168}a^{6}+\frac{751961}{9058792}a^{5}-\frac{3016659}{9058792}a^{4}+\frac{2038875}{4529396}a^{3}+\frac{339543}{1132349}a^{2}+\frac{204647}{2264698}a-\frac{204647}{2264698}$ 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:  $2$

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:  Trivial group, which has order $1$ (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:  $13$
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:   $\frac{42479}{4529396}a^{16}-\frac{453673}{4529396}a^{14}-\frac{329701}{2264698}a^{12}+\frac{2224769}{1132349}a^{10}-\frac{3404795}{4529396}a^{8}-\frac{40830689}{4529396}a^{6}+\frac{19486689}{2264698}a^{4}+\frac{7763789}{2264698}a^{2}-\frac{505789}{1132349}$, $\frac{60021}{9058792}a^{16}-\frac{1256797}{18117584}a^{14}-\frac{1009091}{9058792}a^{12}+\frac{3000943}{2264698}a^{10}-\frac{644794}{1132349}a^{8}-\frac{107705479}{18117584}a^{6}+\frac{8634987}{1132349}a^{4}+\frac{3587541}{2264698}a^{2}-\frac{3473428}{1132349}$, $\frac{322307}{9058792}a^{17}-\frac{5592627}{18117584}a^{15}-\frac{10584577}{9058792}a^{13}+\frac{11598661}{2264698}a^{11}+\frac{31854045}{4529396}a^{9}-\frac{371601869}{18117584}a^{7}-\frac{12504215}{4529396}a^{5}+\frac{42813659}{4529396}a^{3}-\frac{1469591}{1132349}a$, $\frac{34280}{1132349}a^{17}-\frac{312395}{1132349}a^{15}-\frac{4040039}{4529396}a^{13}+\frac{44326373}{9058792}a^{11}+\frac{41287529}{9058792}a^{9}-\frac{195033783}{9058792}a^{7}+\frac{14906557}{9058792}a^{5}+\frac{60677751}{4529396}a^{3}+\frac{1824284}{1132349}a$, $\frac{106941}{9058792}a^{16}-\frac{2032687}{18117584}a^{14}-\frac{2629683}{9058792}a^{12}+\frac{4299803}{2264698}a^{10}+\frac{2938609}{4529396}a^{8}-\frac{123459413}{18117584}a^{6}+\frac{22644065}{4529396}a^{4}-\frac{2395018}{1132349}a^{2}+\frac{491408}{1132349}$, $\frac{1299311}{36235168}a^{17}+\frac{1321235}{36235168}a^{16}-\frac{11835883}{36235168}a^{15}-\frac{11798505}{36235168}a^{14}-\frac{9402311}{9058792}a^{13}-\frac{20064217}{18117584}a^{12}+\frac{101792365}{18117584}a^{11}+\frac{99038619}{18117584}a^{10}+\frac{173968263}{36235168}a^{9}+\frac{202620299}{36235168}a^{8}-\frac{823300571}{36235168}a^{7}-\frac{776896901}{36235168}a^{6}+\frac{95683879}{18117584}a^{5}+\frac{30414403}{9058792}a^{4}+\frac{57679653}{9058792}a^{3}+\frac{5857658}{1132349}a^{2}+\frac{386844}{1132349}a+\frac{543889}{1132349}$, $\frac{1299311}{36235168}a^{17}-\frac{1321235}{36235168}a^{16}-\frac{11835883}{36235168}a^{15}+\frac{11798505}{36235168}a^{14}-\frac{9402311}{9058792}a^{13}+\frac{20064217}{18117584}a^{12}+\frac{101792365}{18117584}a^{11}-\frac{99038619}{18117584}a^{10}+\frac{173968263}{36235168}a^{9}-\frac{202620299}{36235168}a^{8}-\frac{823300571}{36235168}a^{7}+\frac{776896901}{36235168}a^{6}+\frac{95683879}{18117584}a^{5}-\frac{30414403}{9058792}a^{4}+\frac{57679653}{9058792}a^{3}-\frac{5857658}{1132349}a^{2}+\frac{386844}{1132349}a-\frac{543889}{1132349}$, $\frac{49589}{36235168}a^{17}-\frac{56259}{36235168}a^{16}-\frac{71539}{36235168}a^{15}+\frac{209949}{36235168}a^{14}-\frac{1148631}{9058792}a^{13}+\frac{2168219}{18117584}a^{12}-\frac{2932667}{18117584}a^{11}-\frac{189779}{18117584}a^{10}+\frac{56985997}{36235168}a^{9}-\frac{46243259}{36235168}a^{8}+\frac{55049989}{36235168}a^{7}+\frac{2026689}{36235168}a^{6}-\frac{94766747}{18117584}a^{5}+\frac{5374843}{2264698}a^{4}+\frac{57377}{9058792}a^{3}-\frac{377993}{1132349}a^{2}+\frac{2393043}{2264698}a-\frac{321645}{1132349}$, $\frac{49589}{36235168}a^{17}+\frac{56259}{36235168}a^{16}-\frac{71539}{36235168}a^{15}-\frac{209949}{36235168}a^{14}-\frac{1148631}{9058792}a^{13}-\frac{2168219}{18117584}a^{12}-\frac{2932667}{18117584}a^{11}+\frac{189779}{18117584}a^{10}+\frac{56985997}{36235168}a^{9}+\frac{46243259}{36235168}a^{8}+\frac{55049989}{36235168}a^{7}-\frac{2026689}{36235168}a^{6}-\frac{94766747}{18117584}a^{5}-\frac{5374843}{2264698}a^{4}+\frac{57377}{9058792}a^{3}+\frac{377993}{1132349}a^{2}+\frac{2393043}{2264698}a+\frac{321645}{1132349}$, $\frac{47811}{4529396}a^{17}-\frac{330979}{18117584}a^{16}-\frac{705915}{9058792}a^{15}+\frac{3170651}{18117584}a^{14}-\frac{8515111}{18117584}a^{13}+\frac{1042987}{2264698}a^{12}+\frac{10116147}{9058792}a^{11}-\frac{28856519}{9058792}a^{10}+\frac{18916001}{4529396}a^{9}-\frac{27293247}{18117584}a^{8}-\frac{18706323}{4529396}a^{7}+\frac{245241271}{18117584}a^{6}-\frac{165364669}{18117584}a^{5}-\frac{50523131}{9058792}a^{4}+\frac{4777920}{1132349}a^{3}-\frac{15299989}{4529396}a^{2}+\frac{3529428}{1132349}a-\frac{139280}{1132349}$, $\frac{35521}{9058792}a^{17}-\frac{40}{1132349}a^{16}-\frac{756085}{18117584}a^{15}-\frac{69755}{9058792}a^{14}-\frac{1060375}{18117584}a^{13}+\frac{681967}{9058792}a^{12}+\frac{7079289}{9058792}a^{11}+\frac{948593}{4529396}a^{10}-\frac{3398781}{9058792}a^{9}-\frac{5587397}{4529396}a^{8}-\frac{57122903}{18117584}a^{7}-\frac{6756783}{9058792}a^{6}+\frac{71276343}{18117584}a^{5}+\frac{41883483}{9058792}a^{4}-\frac{7114179}{9058792}a^{3}-\frac{6075697}{2264698}a^{2}+\frac{303363}{1132349}a-\frac{375541}{2264698}$, $\frac{1306349}{18117584}a^{17}+\frac{10989}{2264698}a^{16}-\frac{6088581}{9058792}a^{15}-\frac{67331}{1132349}a^{14}-\frac{35361165}{18117584}a^{13}-\frac{158819}{9058792}a^{12}+\frac{106711419}{9058792}a^{11}+\frac{1511967}{1132349}a^{10}+\frac{133160511}{18117584}a^{9}-\frac{5286909}{4529396}a^{8}-\frac{438596481}{9058792}a^{7}-\frac{16175211}{2264698}a^{6}+\frac{361161625}{18117584}a^{5}+\frac{59736121}{9058792}a^{4}+\frac{118394543}{9058792}a^{3}+\frac{5557291}{1132349}a^{2}-\frac{1533352}{1132349}a-\frac{800045}{2264698}$, $\frac{263149}{9058792}a^{17}-\frac{105627}{18117584}a^{16}-\frac{4702693}{18117584}a^{15}+\frac{456765}{9058792}a^{14}-\frac{971682}{1132349}a^{13}+\frac{826063}{4529396}a^{12}+\frac{37844865}{9058792}a^{11}-\frac{3455435}{4529396}a^{10}+\frac{16242519}{4529396}a^{9}-\frac{12659103}{18117584}a^{8}-\frac{264401955}{18117584}a^{7}+\frac{24712489}{9058792}a^{6}+\frac{75057139}{9058792}a^{5}-\frac{23571911}{9058792}a^{4}-\frac{25942279}{9058792}a^{3}-\frac{10042469}{4529396}a^{2}-\frac{4063101}{2264698}a-\frac{544867}{2264698}$ 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:  \( 19355196.3347 \) (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)
 
Unit signature rank:  \( 10 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 19355196.3347 \cdot 1}{2\cdot\sqrt{2855870434202241384378793984}}\cr\approx \mathstrut & 0.28901325769 \end{aligned}\] (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^18 - 10*x^16 - 21*x^14 + 184*x^12 - x^10 - 778*x^8 + 693*x^6 + 140*x^4 - 176*x^2 - 16) 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^18 - 10*x^16 - 21*x^14 + 184*x^12 - x^10 - 778*x^8 + 693*x^6 + 140*x^4 - 176*x^2 - 16, 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^18 - 10*x^16 - 21*x^14 + 184*x^12 - x^10 - 778*x^8 + 693*x^6 + 140*x^4 - 176*x^2 - 16); 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^18 - 10*x^16 - 21*x^14 + 184*x^12 - x^10 - 778*x^8 + 693*x^6 + 140*x^4 - 176*x^2 - 16); 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

$C_2^6:S_3^2$ (as 18T373):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 2304
The 30 conjugacy class representatives for $C_2^6:S_3^2$
Character table for $C_2^6:S_3^2$

Intermediate fields

3.3.148.1, 3.3.404.1, 9.9.3340021539392.1

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

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.4.126546736084484096.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ R ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(2\) Copy content Toggle raw display 2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.6.16a1.8$x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 233 x^{6} + 234 x^{5} + 192 x^{4} + 124 x^{3} + 63 x^{2} + 26 x + 7$$6$$2$$16$12T30$$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.1.2.1a1.1$x^{2} + 37$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.1.2.1a1.1$x^{2} + 37$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
37.2.2.2a1.2$x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(101\) Copy content Toggle raw display 101.3.1.0a1.1$x^{3} + 3 x + 99$$1$$3$$0$$C_3$$$[\ ]^{3}$$
101.3.1.0a1.1$x^{3} + 3 x + 99$$1$$3$$0$$C_3$$$[\ ]^{3}$$
101.6.2.6a1.2$x^{12} + 4 x^{10} + 180 x^{9} + 44 x^{8} + 494 x^{7} + 8184 x^{6} + 3868 x^{5} + 12468 x^{4} + 3040 x^{3} + 4569 x^{2} + 268 x + 105$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$

Spectrum of ring of integers

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