Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 4*x^16 + 2*x^15 - 21*x^14 + 110*x^13 - 211*x^12 - 48*x^11 + 635*x^10 - 600*x^9 - 216*x^8 + 538*x^7 - 180*x^6 + 98*x^5 - 177*x^4 + 46*x^3 + 25*x^2 - 2*x - 1)
gp: K = bnfinit(y^18 - 4*y^17 + 4*y^16 + 2*y^15 - 21*y^14 + 110*y^13 - 211*y^12 - 48*y^11 + 635*y^10 - 600*y^9 - 216*y^8 + 538*y^7 - 180*y^6 + 98*y^5 - 177*y^4 + 46*y^3 + 25*y^2 - 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 4*x^16 + 2*x^15 - 21*x^14 + 110*x^13 - 211*x^12 - 48*x^11 + 635*x^10 - 600*x^9 - 216*x^8 + 538*x^7 - 180*x^6 + 98*x^5 - 177*x^4 + 46*x^3 + 25*x^2 - 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 4*x^16 + 2*x^15 - 21*x^14 + 110*x^13 - 211*x^12 - 48*x^11 + 635*x^10 - 600*x^9 - 216*x^8 + 538*x^7 - 180*x^6 + 98*x^5 - 177*x^4 + 46*x^3 + 25*x^2 - 2*x - 1)
\( x^{18} - 4 x^{17} + 4 x^{16} + 2 x^{15} - 21 x^{14} + 110 x^{13} - 211 x^{12} - 48 x^{11} + 635 x^{10} + \cdots - 1 \)
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: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(60874624883546499579904\)
\(\medspace = 2^{42}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.44\) | 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: | $2^{11/4}7^{2/3}\approx 24.616776431006123$ | ||
| Ramified primes: |
\(2\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{175129166088662}a^{17}+\frac{43650028867233}{175129166088662}a^{16}-\frac{398454063906}{3019468380839}a^{15}-\frac{8264620364508}{87564583044331}a^{14}+\frac{4158498403639}{87564583044331}a^{13}-\frac{40014890261665}{175129166088662}a^{12}-\frac{2989547476943}{87564583044331}a^{11}+\frac{3439768569211}{87564583044331}a^{10}+\frac{65183530632199}{175129166088662}a^{9}-\frac{84583011395289}{175129166088662}a^{8}-\frac{32896502552087}{87564583044331}a^{7}-\frac{2868291696141}{87564583044331}a^{6}-\frac{16874166447783}{175129166088662}a^{5}-\frac{4449807375054}{87564583044331}a^{4}-\frac{24534995258385}{87564583044331}a^{3}-\frac{1585361975730}{87564583044331}a^{2}-\frac{6178194646999}{87564583044331}a+\frac{77600054049475}{175129166088662}$
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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{113110633635}{3019468380839}a^{17}+\frac{7777145038}{3019468380839}a^{16}-\frac{761827404437}{3019468380839}a^{15}+\frac{804616668907}{6038936761678}a^{14}-\frac{1124209980176}{3019468380839}a^{13}+\frac{8598417385339}{6038936761678}a^{12}+\frac{15692521125872}{3019468380839}a^{11}-\frac{48359102149652}{3019468380839}a^{10}-\frac{10638268766545}{3019468380839}a^{9}+\frac{118656564306253}{3019468380839}a^{8}-\frac{51475446798829}{3019468380839}a^{7}-\frac{178429364405605}{6038936761678}a^{6}+\frac{45752302592471}{3019468380839}a^{5}+\frac{39793091272405}{6038936761678}a^{4}+\frac{23841422084581}{3019468380839}a^{3}-\frac{44520176203587}{6038936761678}a^{2}-\frac{11959432414702}{3019468380839}a+\frac{4523357072345}{6038936761678}$, $\frac{873856074628}{3019468380839}a^{17}-\frac{2957398439623}{3019468380839}a^{16}+\frac{1897085181062}{3019468380839}a^{15}+\frac{2389085774926}{3019468380839}a^{14}-\frac{16925488869960}{3019468380839}a^{13}+\frac{172337936489639}{6038936761678}a^{12}-\frac{135024454833196}{3019468380839}a^{11}-\frac{106898899693546}{3019468380839}a^{10}+\frac{473179900483890}{3019468380839}a^{9}-\frac{274562442548986}{3019468380839}a^{8}-\frac{285789062764888}{3019468380839}a^{7}+\frac{300317774484485}{3019468380839}a^{6}-\frac{28673102468994}{3019468380839}a^{5}+\frac{154457334862863}{6038936761678}a^{4}-\frac{106309350465906}{3019468380839}a^{3}-\frac{6520298890633}{3019468380839}a^{2}+\frac{13551396426814}{3019468380839}a+\frac{1433998685919}{6038936761678}$, $a$, $\frac{40630816438475}{175129166088662}a^{17}-\frac{72014963797459}{87564583044331}a^{16}+\frac{1813763166335}{3019468380839}a^{15}+\frac{53341993698551}{87564583044331}a^{14}-\frac{800843332149883}{175129166088662}a^{13}+\frac{20\!\cdots\!48}{87564583044331}a^{12}-\frac{34\!\cdots\!06}{87564583044331}a^{11}-\frac{21\!\cdots\!27}{87564583044331}a^{10}+\frac{23\!\cdots\!75}{175129166088662}a^{9}-\frac{76\!\cdots\!87}{87564583044331}a^{8}-\frac{61\!\cdots\!07}{87564583044331}a^{7}+\frac{78\!\cdots\!26}{87564583044331}a^{6}-\frac{13\!\cdots\!76}{87564583044331}a^{5}+\frac{18\!\cdots\!23}{87564583044331}a^{4}-\frac{28\!\cdots\!15}{87564583044331}a^{3}+\frac{384740709724790}{87564583044331}a^{2}+\frac{717146870831691}{175129166088662}a+\frac{4688045826607}{87564583044331}$, $\frac{13247911621410}{87564583044331}a^{17}-\frac{99028081212893}{175129166088662}a^{16}+\frac{3407522710211}{6038936761678}a^{15}+\frac{16672049634326}{87564583044331}a^{14}-\frac{546119683965033}{175129166088662}a^{13}+\frac{28\!\cdots\!65}{175129166088662}a^{12}-\frac{25\!\cdots\!16}{87564583044331}a^{11}-\frac{531555301791888}{87564583044331}a^{10}+\frac{75\!\cdots\!19}{87564583044331}a^{9}-\frac{15\!\cdots\!81}{175129166088662}a^{8}-\frac{32\!\cdots\!03}{175129166088662}a^{7}+\frac{63\!\cdots\!08}{87564583044331}a^{6}-\frac{55\!\cdots\!97}{175129166088662}a^{5}+\frac{14\!\cdots\!87}{87564583044331}a^{4}-\frac{49\!\cdots\!29}{175129166088662}a^{3}+\frac{892837807252642}{87564583044331}a^{2}+\frac{540886258474301}{175129166088662}a+\frac{54743369837867}{175129166088662}$, $\frac{21782088254357}{175129166088662}a^{17}-\frac{35972170971098}{87564583044331}a^{16}+\frac{2276036599177}{6038936761678}a^{15}+\frac{4201305478026}{87564583044331}a^{14}-\frac{428633534475345}{175129166088662}a^{13}+\frac{21\!\cdots\!59}{175129166088662}a^{12}-\frac{17\!\cdots\!99}{87564583044331}a^{11}-\frac{485584149777107}{87564583044331}a^{10}+\frac{10\!\cdots\!13}{175129166088662}a^{9}-\frac{53\!\cdots\!19}{87564583044331}a^{8}-\frac{256158864921083}{175129166088662}a^{7}+\frac{41\!\cdots\!27}{87564583044331}a^{6}-\frac{29\!\cdots\!82}{87564583044331}a^{5}+\frac{24\!\cdots\!59}{175129166088662}a^{4}-\frac{29\!\cdots\!59}{175129166088662}a^{3}+\frac{854908884185007}{87564583044331}a^{2}+\frac{237618918024773}{175129166088662}a-\frac{25231805327857}{175129166088662}$, $\frac{28038709663107}{175129166088662}a^{17}-\frac{94415452955571}{175129166088662}a^{16}+\frac{288593233041}{6038936761678}a^{15}+\frac{189690231092747}{175129166088662}a^{14}-\frac{521284406689249}{175129166088662}a^{13}+\frac{13\!\cdots\!38}{87564583044331}a^{12}-\frac{17\!\cdots\!30}{87564583044331}a^{11}-\frac{37\!\cdots\!79}{87564583044331}a^{10}+\frac{18\!\cdots\!67}{175129166088662}a^{9}+\frac{956751087798679}{175129166088662}a^{8}-\frac{24\!\cdots\!75}{175129166088662}a^{7}+\frac{77\!\cdots\!79}{175129166088662}a^{6}+\frac{51\!\cdots\!62}{87564583044331}a^{5}+\frac{174133456640493}{175129166088662}a^{4}-\frac{17\!\cdots\!43}{175129166088662}a^{3}-\frac{41\!\cdots\!17}{175129166088662}a^{2}+\frac{12\!\cdots\!25}{175129166088662}a+\frac{211407345552407}{87564583044331}$, $\frac{11125459930270}{87564583044331}a^{17}-\frac{40477629087884}{87564583044331}a^{16}+\frac{599219560009}{3019468380839}a^{15}+\frac{54735277052308}{87564583044331}a^{14}-\frac{210806800780249}{87564583044331}a^{13}+\frac{22\!\cdots\!07}{175129166088662}a^{12}-\frac{17\!\cdots\!47}{87564583044331}a^{11}-\frac{21\!\cdots\!97}{87564583044331}a^{10}+\frac{70\!\cdots\!96}{87564583044331}a^{9}-\frac{20\!\cdots\!89}{87564583044331}a^{8}-\frac{65\!\cdots\!52}{87564583044331}a^{7}+\frac{37\!\cdots\!64}{87564583044331}a^{6}+\frac{11\!\cdots\!67}{87564583044331}a^{5}+\frac{15\!\cdots\!77}{175129166088662}a^{4}-\frac{771896305952959}{87564583044331}a^{3}-\frac{795364086348702}{87564583044331}a^{2}+\frac{453611413679357}{87564583044331}a+\frac{205807394608347}{175129166088662}$, $\frac{12084514779767}{87564583044331}a^{17}-\frac{74243182132197}{175129166088662}a^{16}+\frac{630278500369}{3019468380839}a^{15}+\frac{29615763099823}{87564583044331}a^{14}-\frac{448498350480203}{175129166088662}a^{13}+\frac{11\!\cdots\!66}{87564583044331}a^{12}-\frac{15\!\cdots\!44}{87564583044331}a^{11}-\frac{16\!\cdots\!97}{87564583044331}a^{10}+\frac{57\!\cdots\!28}{87564583044331}a^{9}-\frac{52\!\cdots\!33}{175129166088662}a^{8}-\frac{35\!\cdots\!59}{87564583044331}a^{7}+\frac{28\!\cdots\!89}{87564583044331}a^{6}+\frac{31900705548393}{175129166088662}a^{5}+\frac{24\!\cdots\!31}{175129166088662}a^{4}-\frac{18\!\cdots\!92}{87564583044331}a^{3}+\frac{130017252913319}{87564583044331}a^{2}+\frac{617949944413017}{175129166088662}a+\frac{65250767602970}{87564583044331}$, $\frac{7101249297074}{87564583044331}a^{17}-\frac{13353156607065}{87564583044331}a^{16}-\frac{520930687983}{3019468380839}a^{15}+\frac{31907419296887}{87564583044331}a^{14}-\frac{238088345799305}{175129166088662}a^{13}+\frac{10\!\cdots\!61}{175129166088662}a^{12}-\frac{128110569876210}{87564583044331}a^{11}-\frac{21\!\cdots\!43}{87564583044331}a^{10}+\frac{24\!\cdots\!85}{87564583044331}a^{9}+\frac{20\!\cdots\!43}{87564583044331}a^{8}-\frac{42\!\cdots\!56}{87564583044331}a^{7}+\frac{578452208558797}{87564583044331}a^{6}+\frac{28\!\cdots\!77}{175129166088662}a^{5}-\frac{832976527944985}{175129166088662}a^{4}+\frac{535238937188163}{87564583044331}a^{3}-\frac{970839001771309}{87564583044331}a^{2}+\frac{832042759117949}{175129166088662}a+\frac{8427308659145}{175129166088662}$, $\frac{7902658138491}{87564583044331}a^{17}-\frac{19218785461295}{175129166088662}a^{16}-\frac{1398036994445}{6038936761678}a^{15}+\frac{18978542142449}{175129166088662}a^{14}-\frac{221525003598053}{175129166088662}a^{13}+\frac{995836143127957}{175129166088662}a^{12}+\frac{124185390670897}{87564583044331}a^{11}-\frac{19\!\cdots\!94}{87564583044331}a^{10}+\frac{680822016950187}{87564583044331}a^{9}+\frac{55\!\cdots\!17}{175129166088662}a^{8}-\frac{18\!\cdots\!33}{175129166088662}a^{7}-\frac{46\!\cdots\!71}{175129166088662}a^{6}-\frac{60667473705239}{175129166088662}a^{5}+\frac{11\!\cdots\!71}{87564583044331}a^{4}+\frac{667463303019207}{175129166088662}a^{3}-\frac{165238029876157}{175129166088662}a^{2}-\frac{668845373303625}{175129166088662}a+\frac{108957908934209}{175129166088662}$
| 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: | \( 36150.1579213 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 36150.1579213 \cdot 1}{2\cdot\sqrt{60874624883546499579904}}\cr\approx \mathstrut & 0.288483672596 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 4*x^16 + 2*x^15 - 21*x^14 + 110*x^13 - 211*x^12 - 48*x^11 + 635*x^10 - 600*x^9 - 216*x^8 + 538*x^7 - 180*x^6 + 98*x^5 - 177*x^4 + 46*x^3 + 25*x^2 - 2*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))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^18 - 4*x^17 + 4*x^16 + 2*x^15 - 21*x^14 + 110*x^13 - 211*x^12 - 48*x^11 + 635*x^10 - 600*x^9 - 216*x^8 + 538*x^7 - 180*x^6 + 98*x^5 - 177*x^4 + 46*x^3 + 25*x^2 - 2*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)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 4*x^17 + 4*x^16 + 2*x^15 - 21*x^14 + 110*x^13 - 211*x^12 - 48*x^11 + 635*x^10 - 600*x^9 - 216*x^8 + 538*x^7 - 180*x^6 + 98*x^5 - 177*x^4 + 46*x^3 + 25*x^2 - 2*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)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 4*x^16 + 2*x^15 - 21*x^14 + 110*x^13 - 211*x^12 - 48*x^11 + 635*x^10 - 600*x^9 - 216*x^8 + 538*x^7 - 180*x^6 + 98*x^5 - 177*x^4 + 46*x^3 + 25*x^2 - 2*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
$C_3^2:C_{12}$ (as 18T44):
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 18 conjugacy class representatives for $C_3^2:C_{12}$ |
| Character table for $C_3^2:C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 6.6.1229312.1, 6.2.39337984.1 |
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 siblings: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.20624432955392.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.9.1 | $x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.12.33.344 | $x^{12} + 56 x^{10} - 392 x^{9} + 226 x^{8} - 640 x^{7} - 2480 x^{6} + 3968 x^{5} + 1276 x^{4} - 384 x^{3} + 9280 x^{2} + 24224 x + 31544$ | $4$ | $3$ | $33$ | $C_{12}$ | $[3, 4]^{3}$ | |
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |