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Properties

Label 36.36.900...616.1
Degree $36$
Signature $[36, 0]$
Discriminant $9.007\times 10^{61}$
Root discriminant \(52.60\)
Ramified primes $2,3,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 1)
 
gp: K = bnfinit(y^36 - 36*y^34 + 594*y^32 - 5951*y^30 + 40425*y^28 - 196911*y^26 + 709280*y^24 - 1920270*y^22 + 3932379*y^20 - 6080856*y^18 + 7034958*y^16 - 5982741*y^14 + 3636879*y^12 - 1514853*y^10 + 405648*y^8 - 63358*y^6 + 4956*y^4 - 144*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 1)
 

\( x^{36} - 36 x^{34} + 594 x^{32} - 5951 x^{30} + 40425 x^{28} - 196911 x^{26} + 709280 x^{24} - 1920270 x^{22} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[36, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(90067643300370785938616861622694756230952958181429238736879616\) \(\medspace = 2^{36}\cdot 3^{54}\cdot 7^{30}\) 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:  \(52.60\)
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:  $2\cdot 3^{3/2}7^{5/6}\approx 52.596911665775266$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(5,·)$, $\chi_{252}(11,·)$, $\chi_{252}(17,·)$, $\chi_{252}(19,·)$, $\chi_{252}(23,·)$, $\chi_{252}(25,·)$, $\chi_{252}(155,·)$, $\chi_{252}(31,·)$, $\chi_{252}(37,·)$, $\chi_{252}(41,·)$, $\chi_{252}(95,·)$, $\chi_{252}(71,·)$, $\chi_{252}(173,·)$, $\chi_{252}(179,·)$, $\chi_{252}(55,·)$, $\chi_{252}(185,·)$, $\chi_{252}(187,·)$, $\chi_{252}(191,·)$, $\chi_{252}(193,·)$, $\chi_{252}(139,·)$, $\chi_{252}(199,·)$, $\chi_{252}(205,·)$, $\chi_{252}(209,·)$, $\chi_{252}(85,·)$, $\chi_{252}(89,·)$, $\chi_{252}(223,·)$, $\chi_{252}(101,·)$, $\chi_{252}(103,·)$, $\chi_{252}(107,·)$, $\chi_{252}(109,·)$, $\chi_{252}(239,·)$, $\chi_{252}(115,·)$, $\chi_{252}(169,·)$, $\chi_{252}(121,·)$, $\chi_{252}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (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:  $35$
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:   $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419901a^{12}+277146a^{10}-119394a^{8}+31052a^{6}-4305a^{4}+261a^{2}-4$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461889a^{9}-127899a^{7}+20169a^{5}-1466a^{3}+24a$, $2a^{33}-66a^{31}+990a^{29}-8931a^{27}+54027a^{25}-231336a^{23}+721302a^{21}-1659384a^{19}+2822850a^{17}-3524918a^{15}+3175410a^{13}-2003337a^{11}+844392a^{9}-220671a^{7}+31671a^{5}-2037a^{3}+36a$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+285a^{3}-19a$, $a^{35}-35a^{33}+561a^{31}-5455a^{29}+35930a^{27}-169507a^{25}+590550a^{23}-1543806a^{21}+3046119a^{19}-4525877a^{17}+5012179a^{15}-4059565a^{13}+2333369a^{11}-908962a^{9}+223466a^{7}-30897a^{5}+1965a^{3}-36a$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{35}-35a^{33}+561a^{31}-5456a^{29}+35959a^{27}-169883a^{25}+593424a^{23}-1558182a^{21}+3095499a^{19}-4644741a^{17}+5213067a^{15}-4294592a^{13}+2517697a^{11}-1000493a^{9}+249547a^{7}-34482a^{5}+2145a^{3}-36a$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-1$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520675a^{11}-260327a^{9}+82168a^{7}-14679a^{5}+1185a^{3}-20a$, $a^{2}-2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461889a^{9}-127899a^{7}+20169a^{5}-1466a^{3}+24a-1$, $2a^{33}-66a^{31}+990a^{29}-8931a^{27}+54027a^{25}-231336a^{23}+721302a^{21}-1659384a^{19}+2822850a^{17}-3524918a^{15}+3175410a^{13}-2003337a^{11}+844392a^{9}-220671a^{7}+31671a^{5}-2037a^{3}+36a-1$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168244a^{18}-319752a^{16}-a^{15}+435915a^{14}+15a^{13}-419355a^{12}-90a^{11}+275859a^{10}+275a^{9}-117612a^{8}-450a^{7}+29666a^{6}+378a^{5}-3765a^{4}-140a^{3}+180a^{2}+15a-1$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69767a^{15}-104637a^{13}+107316a^{11}-72655a^{9}+30438a^{7}-6993a^{5}+679a^{3}-12a+1$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361789a^{21}-834879a^{19}+1427490a^{17}-1796867a^{15}+1638561a^{13}-1052505a^{11}+455158a^{9}-123210a^{7}+18495a^{5}-1250a^{3}+24a-1$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}+33a+1$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-136136a^{10}+68068a^{8}+a^{7}-20384a^{6}-7a^{5}+3185a^{4}+14a^{3}-196a^{2}-7a+2$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-a^{14}-3640210a^{13}+14a^{12}+2057510a^{11}-77a^{10}-791350a^{9}+210a^{8}+193800a^{7}-294a^{6}-27132a^{5}+196a^{4}+1785a^{3}-49a^{2}-35a+2$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a-1$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a-1$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7124a^{17}-19363a^{15}+35581a^{13}-43758a^{11}+34815a^{9}-16753a^{7}+4291a^{5}-446a^{3}+8a-1$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a-1$, $a^{23}-23a^{21}+229a^{19}-1292a^{17}+4540a^{15}-10283a^{13}+15015a^{11}-13728a^{9}+7359a^{7}-2035a^{5}+221a^{3}-4a+1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281009a^{13}-243529a^{11}+140933a^{9}-51116a^{7}+10374a^{5}-924a^{3}+16a-1$, $a^{35}-35a^{33}+561a^{31}-5455a^{29}+35930a^{27}-169507a^{25}+590550a^{23}-1543806a^{21}+3046119a^{19}-4525877a^{17}+5012179a^{15}-4059565a^{13}+2333369a^{11}-908962a^{9}+223466a^{7}-30896a^{5}+1960a^{3}-31a-1$, $a^{35}-33a^{33}-a^{32}+494a^{31}+31a^{30}-4435a^{29}-434a^{28}+26594a^{27}+3628a^{26}-112230a^{25}-20176a^{24}+341963a^{23}+78728a^{22}-758725a^{21}-221584a^{20}+1218262a^{19}+455488a^{18}-1381983a^{17}-684948a^{16}+1046806a^{15}+747693a^{14}-454518a^{13}-582165a^{12}+39171a^{11}+313809a^{10}+66706a^{9}-111692a^{8}-34083a^{7}+24358a^{6}+6419a^{5}-2875a^{4}-412a^{3}+140a^{2}+a$, $a+1$, $a^{35}-35a^{33}+561a^{31}-5456a^{29}+35959a^{27}-169883a^{25}+593424a^{23}-1558182a^{21}+3095499a^{19}-4644741a^{17}+5213067a^{15}-4294592a^{13}+2517697a^{11}-1000493a^{9}+249547a^{7}-34482a^{5}+2145a^{3}-35a-1$ 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:  \( 33651165194729250000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{36}\cdot(2\pi)^{0}\cdot 33651165194729250000 \cdot 1}{2\cdot\sqrt{90067643300370785938616861622694756230952958181429238736879616}}\cr\approx \mathstrut & 0.121833173047232 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 36*x^34 + 594*x^32 - 5951*x^30 + 40425*x^28 - 196911*x^26 + 709280*x^24 - 1920270*x^22 + 3932379*x^20 - 6080856*x^18 + 7034958*x^16 - 5982741*x^14 + 3636879*x^12 - 1514853*x^10 + 405648*x^8 - 63358*x^6 + 4956*x^4 - 144*x^2 + 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

$C_6^2$ (as 36T4):

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)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\zeta_{36})^+\), 6.6.3024568512.2, 6.6.4148928.1, 6.6.3024568512.1, 6.6.144027072.1, 6.6.6751269.1, 6.6.7057326528.1, 6.6.330812181.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{21})^+\), 6.6.7057326528.2, 6.6.330812181.2, 9.9.62523502209.1, 12.12.186694177220038656.1, 12.12.448252719505312813056.2, \(\Q(\zeta_{84})^+\), 12.12.448252719505312813056.1, 18.18.27668797159880354103659593728.1, 18.18.351496200956998572502045949952.1, \(\Q(\zeta_{63})^+\)

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 R ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{12}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{12}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.3.0.1}{3} }^{12}$

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 2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
\(3\) Copy content Toggle raw display Deg $18$$6$$3$$27$
Deg $18$$6$$3$$27$
\(7\) Copy content Toggle raw display Deg $36$$6$$6$$30$