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Properties

Label 18.0.227...424.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.279\times 10^{21}$
Root discriminant \(15.37\)
Ramified primes $2,3,23$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^3\wr C_3.S_3^2$ (as 18T734)

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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^18 - 3*x^17 + 6*x^16 - 20*x^15 + 57*x^14 - 102*x^13 + 159*x^12 - 312*x^11 + 567*x^10 - 735*x^9 + 855*x^8 - 1155*x^7 + 1382*x^6 - 1122*x^5 + 567*x^4 - 172*x^3 + 33*x^2 - 6*x + 1)
 
Copy content gp:K = bnfinit(y^18 - 3*y^17 + 6*y^16 - 20*y^15 + 57*y^14 - 102*y^13 + 159*y^12 - 312*y^11 + 567*y^10 - 735*y^9 + 855*y^8 - 1155*y^7 + 1382*y^6 - 1122*y^5 + 567*y^4 - 172*y^3 + 33*y^2 - 6*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 6*x^16 - 20*x^15 + 57*x^14 - 102*x^13 + 159*x^12 - 312*x^11 + 567*x^10 - 735*x^9 + 855*x^8 - 1155*x^7 + 1382*x^6 - 1122*x^5 + 567*x^4 - 172*x^3 + 33*x^2 - 6*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 3*x^17 + 6*x^16 - 20*x^15 + 57*x^14 - 102*x^13 + 159*x^12 - 312*x^11 + 567*x^10 - 735*x^9 + 855*x^8 - 1155*x^7 + 1382*x^6 - 1122*x^5 + 567*x^4 - 172*x^3 + 33*x^2 - 6*x + 1)
 

\( x^{18} - 3 x^{17} + 6 x^{16} - 20 x^{15} + 57 x^{14} - 102 x^{13} + 159 x^{12} - 312 x^{11} + 567 x^{10} + \cdots + 1 \) 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:  $18$
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:  $[0, 9]$
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:   \(-2279403313291670831424\) \(\medspace = -\,2^{6}\cdot 3^{21}\cdot 23^{7}\) 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:  \(15.37\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(23\) 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:  \(\Q(\sqrt{-69}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2330038077427}a^{17}-\frac{890305655253}{2330038077427}a^{16}-\frac{507172759119}{2330038077427}a^{15}+\frac{989445529394}{2330038077427}a^{14}-\frac{603625509136}{2330038077427}a^{13}-\frac{1075849548136}{2330038077427}a^{12}+\frac{25905838358}{2330038077427}a^{11}-\frac{499363146033}{2330038077427}a^{10}-\frac{336148636222}{2330038077427}a^{9}+\frac{1009275826339}{2330038077427}a^{8}-\frac{1155682459554}{2330038077427}a^{7}+\frac{112293079559}{2330038077427}a^{6}-\frac{440544840590}{2330038077427}a^{5}-\frac{769059449896}{2330038077427}a^{4}+\frac{824013542400}{2330038077427}a^{3}+\frac{363420142166}{2330038077427}a^{2}-\frac{185010781437}{2330038077427}a+\frac{207865774968}{2330038077427}$ 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:  $C_{2}$, which has order $2$ (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:  $8$
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:   $\frac{45606205755822}{2330038077427}a^{17}-\frac{108570314278423}{2330038077427}a^{16}+\frac{207081852013542}{2330038077427}a^{15}-\frac{784783368109319}{2330038077427}a^{14}+\frac{21\cdots 17}{2330038077427}a^{13}-\frac{33\cdots 98}{2330038077427}a^{12}+\frac{51\cdots 59}{2330038077427}a^{11}-\frac{11\cdots 97}{2330038077427}a^{10}+\frac{19\cdots 19}{2330038077427}a^{9}-\frac{21\cdots 38}{2330038077427}a^{8}+\frac{25\cdots 54}{2330038077427}a^{7}-\frac{36\cdots 24}{2330038077427}a^{6}+\frac{40\cdots 80}{2330038077427}a^{5}-\frac{26\cdots 39}{2330038077427}a^{4}+\frac{96\cdots 99}{2330038077427}a^{3}-\frac{19\cdots 88}{2330038077427}a^{2}+\frac{317878887335128}{2330038077427}a-\frac{73269471697448}{2330038077427}$, $\frac{20979233420255}{2330038077427}a^{17}-\frac{44943208872073}{2330038077427}a^{16}+\frac{86023499547461}{2330038077427}a^{15}-\frac{343664734815309}{2330038077427}a^{14}+\frac{896830925187906}{2330038077427}a^{13}-\frac{13\cdots 35}{2330038077427}a^{12}+\frac{21\cdots 49}{2330038077427}a^{11}-\frac{46\cdots 16}{2330038077427}a^{10}+\frac{77\cdots 77}{2330038077427}a^{9}-\frac{84\cdots 68}{2330038077427}a^{8}+\frac{10\cdots 75}{2330038077427}a^{7}-\frac{15\cdots 29}{2330038077427}a^{6}+\frac{15\cdots 01}{2330038077427}a^{5}-\frac{94\cdots 80}{2330038077427}a^{4}+\frac{31\cdots 97}{2330038077427}a^{3}-\frac{612198455034958}{2330038077427}a^{2}+\frac{109402253996095}{2330038077427}a-\frac{21871170891314}{2330038077427}$, $\frac{21450046886810}{2330038077427}a^{17}-\frac{53161886127370}{2330038077427}a^{16}+\frac{100035529278509}{2330038077427}a^{15}-\frac{375008795594008}{2330038077427}a^{14}+\frac{10\cdots 31}{2330038077427}a^{13}-\frac{16\cdots 29}{2330038077427}a^{12}+\frac{25\cdots 40}{2330038077427}a^{11}-\frac{53\cdots 02}{2330038077427}a^{10}+\frac{93\cdots 39}{2330038077427}a^{9}-\frac{10\cdots 71}{2330038077427}a^{8}+\frac{12\cdots 46}{2330038077427}a^{7}-\frac{17\cdots 61}{2330038077427}a^{6}+\frac{19\cdots 65}{2330038077427}a^{5}-\frac{13\cdots 47}{2330038077427}a^{4}+\frac{47\cdots 62}{2330038077427}a^{3}-\frac{934450639581130}{2330038077427}a^{2}+\frac{156585506527653}{2330038077427}a-\frac{35256222084568}{2330038077427}$, $\frac{32381438369287}{2330038077427}a^{17}-\frac{77281760939081}{2330038077427}a^{16}+\frac{146999154139884}{2330038077427}a^{15}-\frac{557199663586381}{2330038077427}a^{14}+\frac{15\cdots 49}{2330038077427}a^{13}-\frac{23\cdots 27}{2330038077427}a^{12}+\frac{36\cdots 23}{2330038077427}a^{11}-\frac{78\cdots 62}{2330038077427}a^{10}+\frac{13\cdots 42}{2330038077427}a^{9}-\frac{15\cdots 16}{2330038077427}a^{8}+\frac{18\cdots 24}{2330038077427}a^{7}-\frac{26\cdots 65}{2330038077427}a^{6}+\frac{28\cdots 66}{2330038077427}a^{5}-\frac{18\cdots 16}{2330038077427}a^{4}+\frac{67\cdots 67}{2330038077427}a^{3}-\frac{13\cdots 96}{2330038077427}a^{2}+\frac{236507219186705}{2330038077427}a-\frac{55940654473832}{2330038077427}$, $\frac{48661969044961}{2330038077427}a^{17}-\frac{107303190582679}{2330038077427}a^{16}+\frac{208120587420558}{2330038077427}a^{15}-\frac{810695348221370}{2330038077427}a^{14}+\frac{21\cdots 93}{2330038077427}a^{13}-\frac{32\cdots 99}{2330038077427}a^{12}+\frac{51\cdots 08}{2330038077427}a^{11}-\frac{11\cdots 42}{2330038077427}a^{10}+\frac{18\cdots 89}{2330038077427}a^{9}-\frac{21\cdots 39}{2330038077427}a^{8}+\frac{25\cdots 17}{2330038077427}a^{7}-\frac{36\cdots 07}{2330038077427}a^{6}+\frac{38\cdots 47}{2330038077427}a^{5}-\frac{24\cdots 63}{2330038077427}a^{4}+\frac{88\cdots 33}{2330038077427}a^{3}-\frac{17\cdots 36}{2330038077427}a^{2}+\frac{299301314839599}{2330038077427}a-\frac{66011839338328}{2330038077427}$, $\frac{19482469875663}{2330038077427}a^{17}-\frac{46538931829431}{2330038077427}a^{16}+\frac{89107701115247}{2330038077427}a^{15}-\frac{335844579283705}{2330038077427}a^{14}+\frac{906959851042797}{2330038077427}a^{13}-\frac{14\cdots 67}{2330038077427}a^{12}+\frac{22\cdots 30}{2330038077427}a^{11}-\frac{47\cdots 11}{2330038077427}a^{10}+\frac{81\cdots 83}{2330038077427}a^{9}-\frac{94\cdots 42}{2330038077427}a^{8}+\frac{11\cdots 88}{2330038077427}a^{7}-\frac{15\cdots 31}{2330038077427}a^{6}+\frac{17\cdots 25}{2330038077427}a^{5}-\frac{11\cdots 09}{2330038077427}a^{4}+\frac{41\cdots 59}{2330038077427}a^{3}-\frac{818625931060889}{2330038077427}a^{2}+\frac{128973523113553}{2330038077427}a-\frac{29358061568097}{2330038077427}$, $\frac{15636129207595}{2330038077427}a^{17}-\frac{44256996502862}{2330038077427}a^{16}+\frac{82683188036755}{2330038077427}a^{15}-\frac{292035668111975}{2330038077427}a^{14}+\frac{829221345206712}{2330038077427}a^{13}-\frac{13\cdots 75}{2330038077427}a^{12}+\frac{21\cdots 69}{2330038077427}a^{11}-\frac{43\cdots 07}{2330038077427}a^{10}+\frac{78\cdots 06}{2330038077427}a^{9}-\frac{94\cdots 94}{2330038077427}a^{8}+\frac{10\cdots 57}{2330038077427}a^{7}-\frac{15\cdots 27}{2330038077427}a^{6}+\frac{17\cdots 45}{2330038077427}a^{5}-\frac{12\cdots 94}{2330038077427}a^{4}+\frac{47\cdots 76}{2330038077427}a^{3}-\frac{970988299158366}{2330038077427}a^{2}+\frac{162555435826552}{2330038077427}a-\frac{39066311898769}{2330038077427}$, $\frac{27490718736208}{2330038077427}a^{17}-\frac{67162011181893}{2330038077427}a^{16}+\frac{128729312584916}{2330038077427}a^{15}-\frac{479652654487281}{2330038077427}a^{14}+\frac{13\cdots 71}{2330038077427}a^{13}-\frac{20\cdots 01}{2330038077427}a^{12}+\frac{32\cdots 13}{2330038077427}a^{11}-\frac{68\cdots 56}{2330038077427}a^{10}+\frac{11\cdots 93}{2330038077427}a^{9}-\frac{13\cdots 11}{2330038077427}a^{8}+\frac{16\cdots 02}{2330038077427}a^{7}-\frac{23\cdots 45}{2330038077427}a^{6}+\frac{25\cdots 86}{2330038077427}a^{5}-\frac{17\cdots 00}{2330038077427}a^{4}+\frac{64\cdots 71}{2330038077427}a^{3}-\frac{13\cdots 21}{2330038077427}a^{2}+\frac{222638429375699}{2330038077427}a-\frac{52280041970664}{2330038077427}$ 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:  \( 1034.74957973 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1034.74957973 \cdot 2}{2\cdot\sqrt{2279403313291670831424}}\cr\approx \mathstrut & 0.330783232022 \end{aligned}\] (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 - 3*x^17 + 6*x^16 - 20*x^15 + 57*x^14 - 102*x^13 + 159*x^12 - 312*x^11 + 567*x^10 - 735*x^9 + 855*x^8 - 1155*x^7 + 1382*x^6 - 1122*x^5 + 567*x^4 - 172*x^3 + 33*x^2 - 6*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^18 - 3*x^17 + 6*x^16 - 20*x^15 + 57*x^14 - 102*x^13 + 159*x^12 - 312*x^11 + 567*x^10 - 735*x^9 + 855*x^8 - 1155*x^7 + 1382*x^6 - 1122*x^5 + 567*x^4 - 172*x^3 + 33*x^2 - 6*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^18 - 3*x^17 + 6*x^16 - 20*x^15 + 57*x^14 - 102*x^13 + 159*x^12 - 312*x^11 + 567*x^10 - 735*x^9 + 855*x^8 - 1155*x^7 + 1382*x^6 - 1122*x^5 + 567*x^4 - 172*x^3 + 33*x^2 - 6*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^18 - 3*x^17 + 6*x^16 - 20*x^15 + 57*x^14 - 102*x^13 + 159*x^12 - 312*x^11 + 567*x^10 - 735*x^9 + 855*x^8 - 1155*x^7 + 1382*x^6 - 1122*x^5 + 567*x^4 - 172*x^3 + 33*x^2 - 6*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

$C_2^3\wr C_3.S_3^2$ (as 18T734):

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 55296
The 120 conjugacy class representatives for $C_2^3\wr C_3.S_3^2$
Character table for $C_2^3\wr C_3.S_3^2$

Intermediate fields

3.1.23.1, 9.1.239483061.1

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 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.2.84422344936728549312.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ R ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.3.2.6a2.1$x^{6} + 4 x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 3$$2$$3$$6$$A_4\times C_2$$$[2, 2, 2]^{3}$$
2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(3\) Copy content Toggle raw display 3.3.6.21a8.1$x^{18} + 12 x^{16} + 6 x^{15} + 60 x^{14} + 60 x^{13} + 175 x^{12} + 240 x^{11} + 360 x^{10} + 500 x^{9} + 555 x^{8} + 600 x^{7} + 577 x^{6} + 438 x^{5} + 336 x^{4} + 193 x^{3} + 87 x^{2} + 45 x + 10$$6$$3$$21$18T42not computed
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$$[\ ]$$
23.4.1.0a1.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
23.1.4.3a1.1$x^{4} + 23$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
23.4.2.4a1.2$x^{8} + 6 x^{6} + 38 x^{5} + 19 x^{4} + 114 x^{3} + 391 x^{2} + 190 x + 48$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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