Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3)
gp: K = bnfinit(y^18 - 9*y^17 + 42*y^16 - 129*y^15 + 264*y^14 - 324*y^13 + 126*y^12 + 369*y^11 - 819*y^10 + 711*y^9 + 162*y^8 - 1215*y^7 + 1608*y^6 - 1269*y^5 + 729*y^4 - 333*y^3 + 117*y^2 - 27*y + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3)
\( x^{18} - 9 x^{17} + 42 x^{16} - 129 x^{15} + 264 x^{14} - 324 x^{13} + 126 x^{12} + 369 x^{11} + \cdots + 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-297650820707983690149888\)
\(\medspace = -\,2^{12}\cdot 3^{31}\cdot 7^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.14\) | 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: | $2^{2/3}3^{97/54}7^{1/2}\approx 30.219468835479$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7}a^{15}+\frac{2}{7}a^{14}-\frac{2}{7}a^{13}+\frac{3}{7}a^{12}-\frac{2}{7}a^{10}+\frac{3}{7}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}-\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{16}+\frac{1}{7}a^{14}+\frac{1}{7}a^{12}-\frac{2}{7}a^{11}+\frac{3}{7}a^{9}-\frac{3}{7}a^{8}-\frac{3}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}-\frac{3}{7}a^{4}+\frac{1}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{80802735730283}a^{17}-\frac{4535132339027}{80802735730283}a^{16}-\frac{1793526400509}{80802735730283}a^{15}+\frac{16511630724905}{80802735730283}a^{14}+\frac{35329446934030}{80802735730283}a^{13}+\frac{16534233289659}{80802735730283}a^{12}+\frac{27898233651924}{80802735730283}a^{11}-\frac{11456376131096}{80802735730283}a^{10}-\frac{6305423980162}{80802735730283}a^{9}-\frac{36571844611397}{80802735730283}a^{8}+\frac{12048473729864}{80802735730283}a^{7}+\frac{18216096698679}{80802735730283}a^{6}-\frac{5565846668190}{11543247961469}a^{5}+\frac{37678358602995}{80802735730283}a^{4}-\frac{35618967041375}{80802735730283}a^{3}-\frac{366900269653}{11543247961469}a^{2}-\frac{526136218134}{80802735730283}a+\frac{33305999809806}{80802735730283}$
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
Trivial group, which has order $1$
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{223828226532}{24612469001} a^{17} + \frac{1848054751407}{24612469001} a^{16} - \frac{8004176785874}{24612469001} a^{15} + \frac{22744707214086}{24612469001} a^{14} - \frac{41439062414148}{24612469001} a^{13} + \frac{39693396171237}{24612469001} a^{12} + \frac{4733183177958}{24612469001} a^{11} - \frac{81795320579322}{24612469001} a^{10} + \frac{121021578911702}{24612469001} a^{9} - \frac{61498762820730}{24612469001} a^{8} - \frac{91284537460350}{24612469001} a^{7} + \frac{206828092379994}{24612469001} a^{6} - \frac{195981226422180}{24612469001} a^{5} + \frac{121386158887752}{24612469001} a^{4} - \frac{59494279599486}{24612469001} a^{3} + \frac{22963470724908}{24612469001} a^{2} - \frac{5727667099788}{24612469001} a + \frac{661411624648}{24612469001} \)
(order $6$)
| 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: |
$\frac{380712784373769}{80802735730283}a^{17}-\frac{32\!\cdots\!91}{80802735730283}a^{16}+\frac{14\!\cdots\!06}{80802735730283}a^{15}-\frac{40\!\cdots\!16}{80802735730283}a^{14}+\frac{76\!\cdots\!49}{80802735730283}a^{13}-\frac{78\!\cdots\!81}{80802735730283}a^{12}+\frac{24\!\cdots\!63}{80802735730283}a^{11}+\frac{14\!\cdots\!04}{80802735730283}a^{10}-\frac{22\!\cdots\!80}{80802735730283}a^{9}+\frac{13\!\cdots\!30}{80802735730283}a^{8}+\frac{13\!\cdots\!84}{80802735730283}a^{7}-\frac{37\!\cdots\!19}{80802735730283}a^{6}+\frac{55\!\cdots\!61}{11543247961469}a^{5}-\frac{25\!\cdots\!00}{80802735730283}a^{4}+\frac{13\!\cdots\!84}{80802735730283}a^{3}-\frac{75\!\cdots\!03}{11543247961469}a^{2}+\frac{14\!\cdots\!71}{80802735730283}a-\frac{20\!\cdots\!04}{80802735730283}$, $\frac{91615543400896}{80802735730283}a^{17}-\frac{437021114230461}{80802735730283}a^{16}+\frac{759011713018753}{80802735730283}a^{15}+\frac{11\!\cdots\!01}{80802735730283}a^{14}-\frac{11\!\cdots\!28}{80802735730283}a^{13}+\frac{32\!\cdots\!81}{80802735730283}a^{12}-\frac{41\!\cdots\!35}{80802735730283}a^{11}+\frac{13\!\cdots\!31}{80802735730283}a^{10}+\frac{58\!\cdots\!98}{80802735730283}a^{9}-\frac{10\!\cdots\!24}{80802735730283}a^{8}+\frac{78\!\cdots\!68}{80802735730283}a^{7}+\frac{57\!\cdots\!26}{80802735730283}a^{6}-\frac{32\!\cdots\!42}{1649035423067}a^{5}+\frac{14\!\cdots\!98}{80802735730283}a^{4}-\frac{82\!\cdots\!06}{80802735730283}a^{3}+\frac{56\!\cdots\!71}{11543247961469}a^{2}-\frac{14\!\cdots\!81}{80802735730283}a+\frac{25\!\cdots\!59}{80802735730283}$, $\frac{744879394356091}{80802735730283}a^{17}-\frac{64\!\cdots\!76}{80802735730283}a^{16}+\frac{28\!\cdots\!08}{80802735730283}a^{15}-\frac{84\!\cdots\!04}{80802735730283}a^{14}+\frac{16\!\cdots\!03}{80802735730283}a^{13}-\frac{17\!\cdots\!80}{80802735730283}a^{12}+\frac{21\!\cdots\!31}{80802735730283}a^{11}+\frac{28\!\cdots\!90}{80802735730283}a^{10}-\frac{49\!\cdots\!48}{80802735730283}a^{9}+\frac{32\!\cdots\!56}{80802735730283}a^{8}+\frac{25\!\cdots\!22}{80802735730283}a^{7}-\frac{80\!\cdots\!82}{80802735730283}a^{6}+\frac{12\!\cdots\!78}{11543247961469}a^{5}-\frac{58\!\cdots\!28}{80802735730283}a^{4}+\frac{29\!\cdots\!69}{80802735730283}a^{3}-\frac{17\!\cdots\!35}{11543247961469}a^{2}+\frac{34\!\cdots\!14}{80802735730283}a-\frac{44\!\cdots\!55}{80802735730283}$, $\frac{350244020444764}{80802735730283}a^{17}-\frac{26\!\cdots\!01}{80802735730283}a^{16}+\frac{11\!\cdots\!80}{80802735730283}a^{15}-\frac{29\!\cdots\!74}{80802735730283}a^{14}+\frac{49\!\cdots\!56}{80802735730283}a^{13}-\frac{36\!\cdots\!86}{80802735730283}a^{12}-\frac{24\!\cdots\!50}{80802735730283}a^{11}+\frac{11\!\cdots\!94}{80802735730283}a^{10}-\frac{12\!\cdots\!79}{80802735730283}a^{9}+\frac{31\!\cdots\!28}{80802735730283}a^{8}+\frac{15\!\cdots\!86}{80802735730283}a^{7}-\frac{23\!\cdots\!87}{80802735730283}a^{6}+\frac{26\!\cdots\!49}{11543247961469}a^{5}-\frac{10\!\cdots\!94}{80802735730283}a^{4}+\frac{45\!\cdots\!88}{80802735730283}a^{3}-\frac{20\!\cdots\!40}{11543247961469}a^{2}+\frac{26\!\cdots\!53}{80802735730283}a-\frac{79321765893110}{80802735730283}$, $\frac{229047413567566}{80802735730283}a^{17}-\frac{19\!\cdots\!61}{80802735730283}a^{16}+\frac{89\!\cdots\!63}{80802735730283}a^{15}-\frac{26\!\cdots\!44}{80802735730283}a^{14}+\frac{51\!\cdots\!28}{80802735730283}a^{13}-\frac{56\!\cdots\!73}{80802735730283}a^{12}+\frac{96\!\cdots\!18}{80802735730283}a^{11}+\frac{87\!\cdots\!53}{80802735730283}a^{10}-\frac{15\!\cdots\!46}{80802735730283}a^{9}+\frac{10\!\cdots\!21}{80802735730283}a^{8}+\frac{73\!\cdots\!53}{80802735730283}a^{7}-\frac{25\!\cdots\!63}{80802735730283}a^{6}+\frac{40\!\cdots\!56}{11543247961469}a^{5}-\frac{19\!\cdots\!43}{80802735730283}a^{4}+\frac{99\!\cdots\!43}{80802735730283}a^{3}-\frac{57\!\cdots\!83}{11543247961469}a^{2}+\frac{11\!\cdots\!56}{80802735730283}a-\frac{17\!\cdots\!44}{80802735730283}$, $\frac{412916116558054}{80802735730283}a^{17}-\frac{35\!\cdots\!87}{80802735730283}a^{16}+\frac{15\!\cdots\!42}{80802735730283}a^{15}-\frac{45\!\cdots\!33}{80802735730283}a^{14}+\frac{86\!\cdots\!87}{80802735730283}a^{13}-\frac{88\!\cdots\!65}{80802735730283}a^{12}+\frac{23\!\cdots\!52}{80802735730283}a^{11}+\frac{16\!\cdots\!36}{80802735730283}a^{10}-\frac{26\!\cdots\!41}{80802735730283}a^{9}+\frac{15\!\cdots\!96}{80802735730283}a^{8}+\frac{16\!\cdots\!81}{80802735730283}a^{7}-\frac{43\!\cdots\!39}{80802735730283}a^{6}+\frac{89\!\cdots\!68}{1649035423067}a^{5}-\frac{27\!\cdots\!12}{80802735730283}a^{4}+\frac{13\!\cdots\!51}{80802735730283}a^{3}-\frac{77\!\cdots\!50}{11543247961469}a^{2}+\frac{14\!\cdots\!85}{80802735730283}a-\frac{15\!\cdots\!51}{80802735730283}$, $\frac{107020323136}{80802735730283}a^{17}+\frac{24236693901045}{80802735730283}a^{16}-\frac{191958424616101}{80802735730283}a^{15}+\frac{796160396537890}{80802735730283}a^{14}-\frac{21\!\cdots\!17}{80802735730283}a^{13}+\frac{36\!\cdots\!70}{80802735730283}a^{12}-\frac{27\!\cdots\!11}{80802735730283}a^{11}-\frac{17\!\cdots\!10}{80802735730283}a^{10}+\frac{82\!\cdots\!63}{80802735730283}a^{9}-\frac{96\!\cdots\!06}{80802735730283}a^{8}+\frac{23\!\cdots\!42}{80802735730283}a^{7}+\frac{11\!\cdots\!88}{80802735730283}a^{6}-\frac{363081543049935}{1649035423067}a^{5}+\frac{13\!\cdots\!30}{80802735730283}a^{4}-\frac{73\!\cdots\!51}{80802735730283}a^{3}+\frac{468283999046301}{11543247961469}a^{2}-\frac{960987140257401}{80802735730283}a+\frac{141296255129726}{80802735730283}$, $\frac{103123895999662}{11543247961469}a^{17}-\frac{877044366217624}{11543247961469}a^{16}+\frac{555280242376785}{1649035423067}a^{15}-\frac{11\!\cdots\!61}{11543247961469}a^{14}+\frac{21\!\cdots\!61}{11543247961469}a^{13}-\frac{22\!\cdots\!70}{11543247961469}a^{12}+\frac{664387862604360}{11543247961469}a^{11}+\frac{39\!\cdots\!89}{11543247961469}a^{10}-\frac{64\!\cdots\!27}{11543247961469}a^{9}+\frac{54\!\cdots\!53}{1649035423067}a^{8}+\frac{39\!\cdots\!44}{11543247961469}a^{7}-\frac{15\!\cdots\!24}{1649035423067}a^{6}+\frac{15\!\cdots\!43}{1649035423067}a^{5}-\frac{70\!\cdots\!37}{11543247961469}a^{4}+\frac{35\!\cdots\!68}{11543247961469}a^{3}-\frac{14\!\cdots\!43}{11543247961469}a^{2}+\frac{38\!\cdots\!27}{11543247961469}a-\frac{70749149789827}{1649035423067}$
| 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: | \( 145692.568555 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 145692.568555 \cdot 1}{6\cdot\sqrt{297650820707983690149888}}\cr\approx \mathstrut & 0.679284058462 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3)
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^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3, 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^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3);
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^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3);
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
$C_3\times S_3^2$ (as 18T46):
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 solvable group of order 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.756.1, 6.0.1714608.1, 6.0.314928.1 |
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]
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.729274129765776.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $31$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |