Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169)
gp: K = bnfinit(y^18 - 9*y^17 + 45*y^16 - 140*y^15 + 294*y^14 - 426*y^13 + 498*y^12 - 792*y^11 + 1791*y^10 - 3423*y^9 + 4887*y^8 - 5184*y^7 + 4514*y^6 - 5658*y^5 + 9858*y^4 - 11200*y^3 + 6153*y^2 - 1209*y + 169, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169)
\( x^{18} - 9 x^{17} + 45 x^{16} - 140 x^{15} + 294 x^{14} - 426 x^{13} + 498 x^{12} - 792 x^{11} + \cdots + 169 \)
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: |
\(-593038347905528851673088\)
\(\medspace = -\,2^{12}\cdot 3^{21}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.93\) | 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: | $2^{2/3}3^{7/6}7^{2/3}\approx 20.927956356407396$ | ||
| 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: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{6}+\frac{1}{6}a^{3}+\frac{1}{6}$, $\frac{1}{6}a^{10}-\frac{1}{6}a^{7}+\frac{1}{6}a^{4}+\frac{1}{6}a$, $\frac{1}{6}a^{11}-\frac{1}{6}a^{8}+\frac{1}{6}a^{5}+\frac{1}{6}a^{2}$, $\frac{1}{156}a^{12}+\frac{5}{78}a^{11}+\frac{3}{52}a^{10}+\frac{1}{13}a^{9}-\frac{5}{78}a^{8}+\frac{5}{26}a^{7}+\frac{9}{52}a^{6}+\frac{5}{78}a^{5}-\frac{5}{26}a^{4}-\frac{10}{39}a^{3}-\frac{23}{156}a^{2}+\frac{5}{26}a-\frac{5}{12}$, $\frac{1}{156}a^{13}-\frac{1}{12}a^{11}-\frac{1}{6}a^{8}-\frac{1}{4}a^{7}+\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{4}a^{3}+\frac{1}{6}a^{2}-\frac{53}{156}a$, $\frac{1}{156}a^{14}-\frac{1}{12}a^{10}-\frac{1}{4}a^{8}-\frac{1}{6}a^{7}-\frac{1}{4}a^{6}-\frac{1}{3}a^{5}-\frac{1}{12}a^{4}+\frac{16}{39}a^{2}+\frac{1}{6}a-\frac{1}{4}$, $\frac{1}{468}a^{15}+\frac{1}{468}a^{14}-\frac{1}{468}a^{12}-\frac{23}{468}a^{11}-\frac{11}{234}a^{10}-\frac{25}{468}a^{9}+\frac{23}{468}a^{8}-\frac{17}{468}a^{7}+\frac{5}{26}a^{6}+\frac{9}{52}a^{5}-\frac{139}{468}a^{4}+\frac{5}{18}a^{3}-\frac{121}{468}a^{2}+\frac{35}{468}a+\frac{5}{18}$, $\frac{1}{468}a^{16}-\frac{1}{468}a^{14}-\frac{1}{468}a^{13}-\frac{1}{468}a^{12}-\frac{23}{468}a^{11}+\frac{5}{78}a^{10}-\frac{1}{39}a^{9}-\frac{4}{117}a^{8}-\frac{43}{468}a^{7}-\frac{11}{78}a^{6}+\frac{56}{117}a^{5}-\frac{49}{468}a^{4}+\frac{1}{468}a^{3}-\frac{31}{156}a^{2}+\frac{101}{468}a+\frac{5}{36}$, $\frac{1}{503657647404444}a^{17}-\frac{51864876737}{83942941234074}a^{16}-\frac{105125338103}{251828823702222}a^{15}-\frac{156440431519}{125914411851111}a^{14}+\frac{293203809767}{503657647404444}a^{13}+\frac{1568872944673}{503657647404444}a^{12}+\frac{4041827535765}{55961960822716}a^{11}+\frac{2759241259497}{55961960822716}a^{10}-\frac{7303808137309}{503657647404444}a^{9}+\frac{4237690389965}{125914411851111}a^{8}+\frac{11674571667527}{167885882468148}a^{7}-\frac{65935143995869}{503657647404444}a^{6}-\frac{4910566921159}{125914411851111}a^{5}-\frac{14142041403967}{251828823702222}a^{4}-\frac{6604169349989}{167885882468148}a^{3}-\frac{94718679509359}{503657647404444}a^{2}+\frac{90970548699541}{251828823702222}a+\frac{3963939922063}{12914298651396}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $13$ |
Class group and class number
$C_{3}$, which has order $3$
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: |
\( \frac{46125936002}{41971470617037} a^{17} - \frac{274930706137}{41971470617037} a^{16} + \frac{960672549560}{41971470617037} a^{15} - \frac{1395471863480}{41971470617037} a^{14} - \frac{229311004900}{41971470617037} a^{13} + \frac{4897242058892}{41971470617037} a^{12} - \frac{5863926921544}{41971470617037} a^{11} - \frac{4731915303496}{41971470617037} a^{10} + \frac{12033402039710}{41971470617037} a^{9} + \frac{10056569093210}{41971470617037} a^{8} - \frac{50488038827560}{41971470617037} a^{7} + \frac{97088338790288}{41971470617037} a^{6} - \frac{27320354129460}{13990490205679} a^{5} - \frac{20101946712544}{41971470617037} a^{4} - \frac{33009084563608}{41971470617037} a^{3} + \frac{99087917632720}{13990490205679} a^{2} - \frac{262950778875830}{41971470617037} a + \frac{4210839445855}{3228574662849} \)
(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: |
$\frac{64488651822}{13990490205679}a^{17}-\frac{1989856004715}{55961960822716}a^{16}+\frac{2248665134362}{13990490205679}a^{15}-\frac{12102893968911}{27980980411358}a^{14}+\frac{21631848335751}{27980980411358}a^{13}-\frac{50879623790487}{55961960822716}a^{12}+\frac{14277208706064}{13990490205679}a^{11}-\frac{123972586694523}{55961960822716}a^{10}+\frac{145963372920065}{27980980411358}a^{9}-\frac{487399157059185}{55961960822716}a^{8}+\frac{149025897749484}{13990490205679}a^{7}-\frac{517216465342159}{55961960822716}a^{6}+\frac{215471200676865}{27980980411358}a^{5}-\frac{422394966821523}{27980980411358}a^{4}+\frac{344814176567480}{13990490205679}a^{3}-\frac{997827517288677}{55961960822716}a^{2}+\frac{3886024210053}{1076191554283}a-\frac{7306148451}{2152383108566}$, $\frac{3313586259}{13990490205679}a^{17}-\frac{131968557315}{27980980411358}a^{16}+\frac{387942568526}{13990490205679}a^{15}-\frac{1441186035891}{13990490205679}a^{14}+\frac{3172411985511}{13990490205679}a^{13}-\frac{4593242222055}{13990490205679}a^{12}+\frac{4331288605896}{13990490205679}a^{11}-\frac{12579270356943}{27980980411358}a^{10}+\frac{36062767209273}{27980980411358}a^{9}-\frac{73060003190457}{27980980411358}a^{8}+\frac{50216080484520}{13990490205679}a^{7}-\frac{50911465913327}{13990490205679}a^{6}+\frac{32334908456301}{13990490205679}a^{5}-\frac{40262797539507}{13990490205679}a^{4}+\frac{17201771635091}{2152383108566}a^{3}-\frac{247367961220161}{27980980411358}a^{2}+\frac{1969332050433}{1076191554283}a+\frac{2078979803494}{1076191554283}$, $\frac{5929325225}{55961960822716}a^{17}-\frac{312605767835}{167885882468148}a^{16}+\frac{6568264453981}{503657647404444}a^{15}-\frac{6669766054232}{125914411851111}a^{14}+\frac{1953160135608}{13990490205679}a^{13}-\frac{28888695736240}{125914411851111}a^{12}+\frac{125955648291367}{503657647404444}a^{11}-\frac{59104324965059}{251828823702222}a^{10}+\frac{81429066549131}{125914411851111}a^{9}-\frac{829193424047857}{503657647404444}a^{8}+\frac{13\!\cdots\!17}{503657647404444}a^{7}-\frac{122535046041673}{41971470617037}a^{6}+\frac{357309622536301}{167885882468148}a^{5}-\frac{330463780779413}{251828823702222}a^{4}+\frac{10\!\cdots\!63}{251828823702222}a^{3}-\frac{10\!\cdots\!91}{125914411851111}a^{2}+\frac{13\!\cdots\!93}{251828823702222}a+\frac{15269580125995}{38742895954188}$, $\frac{198441479903}{55961960822716}a^{17}-\frac{1353577962975}{55961960822716}a^{16}+\frac{52301310394249}{503657647404444}a^{15}-\frac{63249068536423}{251828823702222}a^{14}+\frac{11646495254827}{27980980411358}a^{13}-\frac{16562883925129}{38742895954188}a^{12}+\frac{21598856256937}{38742895954188}a^{11}-\frac{52101345944071}{38742895954188}a^{10}+\frac{29515571328179}{9685723988547}a^{9}-\frac{176309375059111}{38742895954188}a^{8}+\frac{204891036255025}{38742895954188}a^{7}-\frac{48727612815907}{12914298651396}a^{6}+\frac{227689087511479}{55961960822716}a^{5}-\frac{10\!\cdots\!49}{125914411851111}a^{4}+\frac{31\!\cdots\!39}{251828823702222}a^{3}-\frac{28\!\cdots\!33}{503657647404444}a^{2}+\frac{90911187815573}{125914411851111}a+\frac{8774124281932}{9685723988547}$, $\frac{26547039349}{19371447977094}a^{17}-\frac{1794628506661}{167885882468148}a^{16}+\frac{6046149595283}{125914411851111}a^{15}-\frac{16249303842427}{125914411851111}a^{14}+\frac{28531835597617}{125914411851111}a^{13}-\frac{132981067919663}{503657647404444}a^{12}+\frac{73203899749997}{251828823702222}a^{11}-\frac{334984936807747}{503657647404444}a^{10}+\frac{64932579438374}{41971470617037}a^{9}-\frac{142969944290243}{55961960822716}a^{8}+\frac{382622412786781}{125914411851111}a^{7}-\frac{12\!\cdots\!23}{503657647404444}a^{6}+\frac{258188622796099}{125914411851111}a^{5}-\frac{11\!\cdots\!99}{251828823702222}a^{4}+\frac{939784846860394}{125914411851111}a^{3}-\frac{791928714044371}{167885882468148}a^{2}-\frac{9849306834316}{41971470617037}a-\frac{4851010007017}{19371447977094}$, $\frac{882531054545}{251828823702222}a^{17}-\frac{76321191043}{3228574662849}a^{16}+\frac{17006134504715}{167885882468148}a^{15}-\frac{13852259907171}{55961960822716}a^{14}+\frac{214655758630673}{503657647404444}a^{13}-\frac{27893942294847}{55961960822716}a^{12}+\frac{87312456058462}{125914411851111}a^{11}-\frac{373349354576927}{251828823702222}a^{10}+\frac{15\!\cdots\!47}{503657647404444}a^{9}-\frac{23\!\cdots\!11}{503657647404444}a^{8}+\frac{15\!\cdots\!91}{251828823702222}a^{7}-\frac{13\!\cdots\!55}{251828823702222}a^{6}+\frac{30\!\cdots\!19}{503657647404444}a^{5}-\frac{17\!\cdots\!99}{167885882468148}a^{4}+\frac{61\!\cdots\!93}{503657647404444}a^{3}-\frac{44\!\cdots\!81}{503657647404444}a^{2}+\frac{18\!\cdots\!73}{251828823702222}a-\frac{17018913327697}{19371447977094}$, $\frac{324707448694}{125914411851111}a^{17}-\frac{7500888965159}{503657647404444}a^{16}+\frac{29681391216875}{503657647404444}a^{15}-\frac{29212209393187}{251828823702222}a^{14}+\frac{20211008698897}{125914411851111}a^{13}-\frac{8566309559395}{83942941234074}a^{12}+\frac{109080008243221}{503657647404444}a^{11}-\frac{55479765687607}{83942941234074}a^{10}+\frac{678832228290689}{503657647404444}a^{9}-\frac{851299015685939}{503657647404444}a^{8}+\frac{783873420894401}{503657647404444}a^{7}-\frac{27020615184925}{125914411851111}a^{6}+\frac{699681412438159}{503657647404444}a^{5}-\frac{199868749854682}{41971470617037}a^{4}+\frac{430645586733982}{125914411851111}a^{3}+\frac{224039284161167}{125914411851111}a^{2}-\frac{286696688852177}{503657647404444}a-\frac{7601664583363}{38742895954188}$, $\frac{85141647428}{125914411851111}a^{17}-\frac{2833025610041}{503657647404444}a^{16}+\frac{772128001279}{27980980411358}a^{15}-\frac{21200630853401}{251828823702222}a^{14}+\frac{30296481580237}{167885882468148}a^{13}-\frac{139106056362967}{503657647404444}a^{12}+\frac{19939102154393}{55961960822716}a^{11}-\frac{278277288433397}{503657647404444}a^{10}+\frac{140254971264536}{125914411851111}a^{9}-\frac{10\!\cdots\!11}{503657647404444}a^{8}+\frac{16\!\cdots\!45}{503657647404444}a^{7}-\frac{18\!\cdots\!55}{503657647404444}a^{6}+\frac{296501203988605}{83942941234074}a^{5}-\frac{10\!\cdots\!69}{251828823702222}a^{4}+\frac{998819723956163}{167885882468148}a^{3}-\frac{35\!\cdots\!39}{503657647404444}a^{2}+\frac{224458999629751}{38742895954188}a-\frac{42364589905243}{19371447977094}$
| 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: | \( 51834.4889774 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 51834.4889774 \cdot 3}{6\cdot\sqrt{593038347905528851673088}}\cr\approx \mathstrut & 0.513648769710 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169)
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 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169, 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 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169);
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 + 45*x^16 - 140*x^15 + 294*x^14 - 426*x^13 + 498*x^12 - 792*x^11 + 1791*x^10 - 3423*x^9 + 4887*x^8 - 5184*x^7 + 4514*x^6 - 5658*x^5 + 9858*x^4 - 11200*x^3 + 6153*x^2 - 1209*x + 169);
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
$C_3\times S_3$ (as 18T3):
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 18 |
| The 9 conjugacy class representatives for $S_3 \times C_3$ |
| Character table for $S_3 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.5292.1 x3, \(\Q(\zeta_{7})^+\), 6.0.84015792.1, 6.0.1714608.2 x2, 6.0.64827.1, 9.3.148203857088.5 x3 |
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 6 sibling: | 6.0.1714608.2 |
| Degree 9 sibling: | 9.3.148203857088.5 |
| Minimal sibling: | 6.0.1714608.2 |
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 | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.1.0.1}{1} }^{18}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 $18$ | $6$ | $3$ | $21$ | |||
|
\(7\)
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |