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Properties

Label 12.12.712...000.1
Degree $12$
Signature $[12, 0]$
Discriminant $7.124\times 10^{25}$
Root discriminant \(142.69\)
Ramified primes $2,5,7,17$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 512*x^10 + 1724*x^9 + 103859*x^8 - 281572*x^7 - 10644338*x^6 + 21700544*x^5 + 578712080*x^4 - 784199124*x^3 - 15736799340*x^2 + 10519199812*x + 166108823761)
 
gp: K = bnfinit(y^12 - 4*y^11 - 512*y^10 + 1724*y^9 + 103859*y^8 - 281572*y^7 - 10644338*y^6 + 21700544*y^5 + 578712080*y^4 - 784199124*y^3 - 15736799340*y^2 + 10519199812*y + 166108823761, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 512*x^10 + 1724*x^9 + 103859*x^8 - 281572*x^7 - 10644338*x^6 + 21700544*x^5 + 578712080*x^4 - 784199124*x^3 - 15736799340*x^2 + 10519199812*x + 166108823761);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 512*x^10 + 1724*x^9 + 103859*x^8 - 281572*x^7 - 10644338*x^6 + 21700544*x^5 + 578712080*x^4 - 784199124*x^3 - 15736799340*x^2 + 10519199812*x + 166108823761)
 

\( x^{12} - 4 x^{11} - 512 x^{10} + 1724 x^{9} + 103859 x^{8} - 281572 x^{7} - 10644338 x^{6} + \cdots + 166108823761 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[12, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(71243920337289728000000000\) \(\medspace = 2^{18}\cdot 5^{9}\cdot 7^{8}\cdot 17^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(142.69\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}5^{3/4}7^{2/3}17^{1/2}\approx 142.69069418206897$
Ramified primes:   \(2\), \(5\), \(7\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4760=2^{3}\cdot 5\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4760}(1563,·)$, $\chi_{4760}(1,·)$, $\chi_{4760}(67,·)$, $\chi_{4760}(1089,·)$, $\chi_{4760}(4489,·)$, $\chi_{4760}(2923,·)$, $\chi_{4760}(681,·)$, $\chi_{4760}(883,·)$, $\chi_{4760}(2787,·)$, $\chi_{4760}(2041,·)$, $\chi_{4760}(3809,·)$, $\chi_{4760}(3467,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{34}a^{6}-\frac{1}{17}a^{5}-\frac{3}{34}a^{4}+\frac{3}{17}a^{3}+\frac{1}{17}a^{2}-\frac{2}{17}a+\frac{1}{34}$, $\frac{1}{34}a^{7}-\frac{7}{34}a^{5}+\frac{7}{17}a^{3}-\frac{7}{34}a+\frac{1}{17}$, $\frac{1}{34}a^{8}-\frac{7}{17}a^{5}-\frac{7}{34}a^{4}+\frac{4}{17}a^{3}+\frac{7}{34}a^{2}+\frac{4}{17}a+\frac{7}{34}$, $\frac{1}{34}a^{9}-\frac{1}{34}a^{5}-\frac{11}{34}a^{3}+\frac{1}{17}a^{2}-\frac{15}{34}a+\frac{7}{17}$, $\frac{1}{2284464069154}a^{10}-\frac{6835828908}{1142232034577}a^{9}-\frac{1023632705}{1142232034577}a^{8}-\frac{23973522013}{2284464069154}a^{7}+\frac{8169254499}{1142232034577}a^{6}-\frac{606723253077}{2284464069154}a^{5}+\frac{201818572927}{1142232034577}a^{4}+\frac{530099055905}{1142232034577}a^{3}+\frac{1135441172327}{2284464069154}a^{2}+\frac{1119593369553}{2284464069154}a+\frac{986520148179}{2284464069154}$, $\frac{1}{16\!\cdots\!14}a^{11}-\frac{38966485463362}{81\!\cdots\!57}a^{10}-\frac{91\!\cdots\!59}{95\!\cdots\!42}a^{9}+\frac{63\!\cdots\!08}{81\!\cdots\!57}a^{8}+\frac{19\!\cdots\!37}{16\!\cdots\!14}a^{7}+\frac{14\!\cdots\!97}{16\!\cdots\!14}a^{6}+\frac{65\!\cdots\!84}{81\!\cdots\!57}a^{5}-\frac{48\!\cdots\!47}{16\!\cdots\!14}a^{4}+\frac{36\!\cdots\!36}{81\!\cdots\!57}a^{3}+\frac{31\!\cdots\!85}{81\!\cdots\!57}a^{2}+\frac{17\!\cdots\!65}{16\!\cdots\!14}a-\frac{27\!\cdots\!93}{68\!\cdots\!26}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{14}{1142232034577}a^{11}+\frac{14925}{1142232034577}a^{10}-\frac{56715}{1142232034577}a^{9}-\frac{12888435}{2284464069154}a^{8}+\frac{18617800}{1142232034577}a^{7}+\frac{1027042051}{1142232034577}a^{6}-\frac{2186714138}{1142232034577}a^{5}-\frac{148881072825}{2284464069154}a^{4}+\frac{104317889180}{1142232034577}a^{3}+\frac{4901710840301}{2284464069154}a^{2}-\frac{1678580195734}{1142232034577}a-\frac{3670084781255}{134380239362}$, $\frac{48\!\cdots\!40}{47\!\cdots\!21}a^{11}-\frac{59\!\cdots\!26}{47\!\cdots\!21}a^{10}-\frac{19\!\cdots\!60}{47\!\cdots\!21}a^{9}+\frac{27\!\cdots\!50}{47\!\cdots\!21}a^{8}+\frac{26\!\cdots\!20}{47\!\cdots\!21}a^{7}-\frac{49\!\cdots\!30}{47\!\cdots\!21}a^{6}-\frac{14\!\cdots\!80}{47\!\cdots\!21}a^{5}+\frac{40\!\cdots\!95}{47\!\cdots\!21}a^{4}+\frac{20\!\cdots\!40}{47\!\cdots\!21}a^{3}-\frac{15\!\cdots\!60}{47\!\cdots\!21}a^{2}+\frac{25\!\cdots\!20}{47\!\cdots\!21}a+\frac{87\!\cdots\!89}{20\!\cdots\!39}$, $\frac{77\!\cdots\!60}{47\!\cdots\!21}a^{11}+\frac{14\!\cdots\!30}{47\!\cdots\!21}a^{10}-\frac{34\!\cdots\!00}{47\!\cdots\!21}a^{9}-\frac{11\!\cdots\!65}{47\!\cdots\!21}a^{8}+\frac{57\!\cdots\!80}{47\!\cdots\!21}a^{7}+\frac{29\!\cdots\!70}{47\!\cdots\!21}a^{6}-\frac{44\!\cdots\!44}{47\!\cdots\!21}a^{5}-\frac{31\!\cdots\!75}{47\!\cdots\!21}a^{4}+\frac{16\!\cdots\!60}{47\!\cdots\!21}a^{3}+\frac{13\!\cdots\!20}{47\!\cdots\!21}a^{2}-\frac{21\!\cdots\!00}{47\!\cdots\!21}a-\frac{86\!\cdots\!35}{20\!\cdots\!39}$, $\frac{13\!\cdots\!94}{81\!\cdots\!57}a^{11}+\frac{13\!\cdots\!35}{81\!\cdots\!57}a^{10}-\frac{62\!\cdots\!15}{81\!\cdots\!57}a^{9}-\frac{13\!\cdots\!45}{16\!\cdots\!14}a^{8}+\frac{11\!\cdots\!60}{81\!\cdots\!57}a^{7}+\frac{12\!\cdots\!81}{81\!\cdots\!57}a^{6}-\frac{91\!\cdots\!06}{81\!\cdots\!57}a^{5}-\frac{21\!\cdots\!75}{16\!\cdots\!14}a^{4}+\frac{34\!\cdots\!00}{81\!\cdots\!57}a^{3}+\frac{81\!\cdots\!21}{16\!\cdots\!14}a^{2}-\frac{47\!\cdots\!94}{81\!\cdots\!57}a-\frac{28\!\cdots\!15}{40\!\cdots\!78}$, $\frac{21\!\cdots\!26}{81\!\cdots\!57}a^{11}-\frac{18\!\cdots\!57}{81\!\cdots\!57}a^{10}-\frac{87\!\cdots\!05}{81\!\cdots\!57}a^{9}+\frac{14\!\cdots\!25}{16\!\cdots\!14}a^{8}+\frac{12\!\cdots\!00}{81\!\cdots\!57}a^{7}-\frac{10\!\cdots\!11}{81\!\cdots\!57}a^{6}-\frac{84\!\cdots\!50}{81\!\cdots\!57}a^{5}+\frac{13\!\cdots\!05}{16\!\cdots\!14}a^{4}+\frac{23\!\cdots\!20}{81\!\cdots\!57}a^{3}-\frac{40\!\cdots\!01}{16\!\cdots\!14}a^{2}-\frac{19\!\cdots\!66}{81\!\cdots\!57}a+\frac{11\!\cdots\!53}{40\!\cdots\!78}$, $\frac{40\!\cdots\!29}{81\!\cdots\!57}a^{11}-\frac{10\!\cdots\!63}{16\!\cdots\!14}a^{10}-\frac{32\!\cdots\!25}{16\!\cdots\!14}a^{9}+\frac{41\!\cdots\!55}{16\!\cdots\!14}a^{8}+\frac{23\!\cdots\!05}{81\!\cdots\!57}a^{7}-\frac{31\!\cdots\!39}{81\!\cdots\!57}a^{6}-\frac{30\!\cdots\!43}{16\!\cdots\!14}a^{5}+\frac{42\!\cdots\!35}{16\!\cdots\!14}a^{4}+\frac{88\!\cdots\!05}{16\!\cdots\!14}a^{3}-\frac{65\!\cdots\!62}{81\!\cdots\!57}a^{2}-\frac{11\!\cdots\!15}{16\!\cdots\!14}a+\frac{33\!\cdots\!19}{34\!\cdots\!63}$, $\frac{38\!\cdots\!11}{81\!\cdots\!57}a^{11}-\frac{10\!\cdots\!13}{16\!\cdots\!14}a^{10}-\frac{30\!\cdots\!99}{16\!\cdots\!14}a^{9}+\frac{41\!\cdots\!21}{16\!\cdots\!14}a^{8}+\frac{22\!\cdots\!03}{81\!\cdots\!57}a^{7}-\frac{30\!\cdots\!77}{81\!\cdots\!57}a^{6}-\frac{27\!\cdots\!17}{16\!\cdots\!14}a^{5}+\frac{42\!\cdots\!45}{16\!\cdots\!14}a^{4}+\frac{74\!\cdots\!07}{16\!\cdots\!14}a^{3}-\frac{65\!\cdots\!20}{81\!\cdots\!57}a^{2}-\frac{67\!\cdots\!53}{16\!\cdots\!14}a+\frac{30\!\cdots\!17}{34\!\cdots\!63}$, $\frac{66\!\cdots\!17}{16\!\cdots\!14}a^{11}-\frac{59\!\cdots\!69}{16\!\cdots\!14}a^{10}-\frac{18\!\cdots\!91}{95\!\cdots\!42}a^{9}+\frac{13\!\cdots\!02}{81\!\cdots\!57}a^{8}+\frac{55\!\cdots\!35}{16\!\cdots\!14}a^{7}-\frac{22\!\cdots\!01}{81\!\cdots\!57}a^{6}-\frac{23\!\cdots\!68}{81\!\cdots\!57}a^{5}+\frac{18\!\cdots\!15}{81\!\cdots\!57}a^{4}+\frac{95\!\cdots\!90}{81\!\cdots\!57}a^{3}-\frac{14\!\cdots\!31}{16\!\cdots\!14}a^{2}-\frac{29\!\cdots\!79}{16\!\cdots\!14}a+\frac{89\!\cdots\!83}{68\!\cdots\!26}$, $\frac{13\!\cdots\!93}{16\!\cdots\!14}a^{11}-\frac{13\!\cdots\!65}{16\!\cdots\!14}a^{10}-\frac{29\!\cdots\!37}{81\!\cdots\!57}a^{9}+\frac{29\!\cdots\!44}{81\!\cdots\!57}a^{8}+\frac{97\!\cdots\!11}{16\!\cdots\!14}a^{7}-\frac{97\!\cdots\!73}{16\!\cdots\!14}a^{6}-\frac{44\!\cdots\!09}{95\!\cdots\!42}a^{5}+\frac{75\!\cdots\!05}{16\!\cdots\!14}a^{4}+\frac{26\!\cdots\!93}{16\!\cdots\!14}a^{3}-\frac{26\!\cdots\!43}{16\!\cdots\!14}a^{2}-\frac{15\!\cdots\!79}{81\!\cdots\!57}a+\frac{69\!\cdots\!84}{34\!\cdots\!63}$, $\frac{58\!\cdots\!67}{16\!\cdots\!14}a^{11}+\frac{10\!\cdots\!76}{81\!\cdots\!57}a^{10}-\frac{14\!\cdots\!11}{81\!\cdots\!57}a^{9}-\frac{11\!\cdots\!61}{16\!\cdots\!14}a^{8}+\frac{26\!\cdots\!84}{81\!\cdots\!57}a^{7}+\frac{11\!\cdots\!27}{81\!\cdots\!57}a^{6}-\frac{23\!\cdots\!89}{81\!\cdots\!57}a^{5}-\frac{21\!\cdots\!57}{16\!\cdots\!14}a^{4}+\frac{18\!\cdots\!41}{16\!\cdots\!14}a^{3}+\frac{91\!\cdots\!79}{16\!\cdots\!14}a^{2}-\frac{26\!\cdots\!15}{16\!\cdots\!14}a-\frac{33\!\cdots\!27}{40\!\cdots\!78}$, $\frac{14\!\cdots\!36}{81\!\cdots\!57}a^{11}+\frac{10\!\cdots\!45}{81\!\cdots\!57}a^{10}-\frac{62\!\cdots\!91}{81\!\cdots\!57}a^{9}-\frac{85\!\cdots\!43}{16\!\cdots\!14}a^{8}+\frac{98\!\cdots\!93}{81\!\cdots\!57}a^{7}+\frac{66\!\cdots\!35}{81\!\cdots\!57}a^{6}-\frac{72\!\cdots\!81}{81\!\cdots\!57}a^{5}-\frac{96\!\cdots\!79}{16\!\cdots\!14}a^{4}+\frac{24\!\cdots\!35}{81\!\cdots\!57}a^{3}+\frac{32\!\cdots\!55}{16\!\cdots\!14}a^{2}-\frac{31\!\cdots\!71}{81\!\cdots\!57}a-\frac{16\!\cdots\!73}{68\!\cdots\!26}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 320040330.009 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 320040330.009 \cdot 2}{2\cdot\sqrt{71243920337289728000000000}}\cr\approx \mathstrut & 0.155306900638 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 512*x^10 + 1724*x^9 + 103859*x^8 - 281572*x^7 - 10644338*x^6 + 21700544*x^5 + 578712080*x^4 - 784199124*x^3 - 15736799340*x^2 + 10519199812*x + 166108823761)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^12 - 4*x^11 - 512*x^10 + 1724*x^9 + 103859*x^8 - 281572*x^7 - 10644338*x^6 + 21700544*x^5 + 578712080*x^4 - 784199124*x^3 - 15736799340*x^2 + 10519199812*x + 166108823761, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^12 - 4*x^11 - 512*x^10 + 1724*x^9 + 103859*x^8 - 281572*x^7 - 10644338*x^6 + 21700544*x^5 + 578712080*x^4 - 784199124*x^3 - 15736799340*x^2 + 10519199812*x + 166108823761);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 512*x^10 + 1724*x^9 + 103859*x^8 - 281572*x^7 - 10644338*x^6 + 21700544*x^5 + 578712080*x^4 - 784199124*x^3 - 15736799340*x^2 + 10519199812*x + 166108823761);
 
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

$C_{12}$ (as 12T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.4.2312000.1, 6.6.300125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-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 ${\href{/padicField/3.12.0.1}{12} }$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}$ R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 2.12.18.27$x^{12} + 24 x^{11} + 300 x^{10} + 2480 x^{9} + 15084 x^{8} + 70848 x^{7} + 263968 x^{6} + 785280 x^{5} + 1858672 x^{4} + 3423104 x^{3} + 4742336 x^{2} + 4511488 x + 2639680$$2$$6$$18$$C_{12}$$[3]^{6}$
\(5\) Copy content Toggle raw display 5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) Copy content Toggle raw display 7.12.8.1$x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
\(17\) Copy content Toggle raw display 17.12.6.2$x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$