Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 3*x^16 - 18*x^14 + 37*x^13 - 27*x^12 - 12*x^11 + 107*x^10 - 158*x^9 - 7*x^8 + 207*x^7 - 134*x^6 - 64*x^5 + 90*x^4 - 10*x^3 - 16*x^2 + 2*x + 1)
gp: K = bnfinit(y^18 - 3*y^17 + 3*y^16 - 18*y^14 + 37*y^13 - 27*y^12 - 12*y^11 + 107*y^10 - 158*y^9 - 7*y^8 + 207*y^7 - 134*y^6 - 64*y^5 + 90*y^4 - 10*y^3 - 16*y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 3*x^16 - 18*x^14 + 37*x^13 - 27*x^12 - 12*x^11 + 107*x^10 - 158*x^9 - 7*x^8 + 207*x^7 - 134*x^6 - 64*x^5 + 90*x^4 - 10*x^3 - 16*x^2 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 3*x^16 - 18*x^14 + 37*x^13 - 27*x^12 - 12*x^11 + 107*x^10 - 158*x^9 - 7*x^8 + 207*x^7 - 134*x^6 - 64*x^5 + 90*x^4 - 10*x^3 - 16*x^2 + 2*x + 1)
\( x^{18} - 3 x^{17} + 3 x^{16} - 18 x^{14} + 37 x^{13} - 27 x^{12} - 12 x^{11} + 107 x^{10} - 158 x^{9} + \cdots + 1 \)
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: | $[10, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(66983486784702356675569\)
\(\medspace = 7^{12}\cdot 97^{2}\cdot 22679^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.54\) | 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: | $7^{2/3}97^{1/2}22679^{1/2}\approx 5427.458494884186$ | ||
| Ramified primes: |
\(7\), \(97\), \(22679\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{16953699937}a^{17}-\frac{1220100652}{16953699937}a^{16}+\frac{1313934667}{16953699937}a^{15}+\frac{1557455767}{16953699937}a^{14}+\frac{3602262090}{16953699937}a^{13}+\frac{5493715858}{16953699937}a^{12}+\frac{123862523}{16953699937}a^{11}-\frac{5979057423}{16953699937}a^{10}-\frac{3699097747}{16953699937}a^{9}-\frac{7365831618}{16953699937}a^{8}+\frac{753846989}{16953699937}a^{7}-\frac{4533560755}{16953699937}a^{6}+\frac{631830468}{16953699937}a^{5}-\frac{7756128463}{16953699937}a^{4}-\frac{6030734522}{16953699937}a^{3}+\frac{7467338025}{16953699937}a^{2}-\frac{1198624817}{16953699937}a-\frac{6645651458}{16953699937}$
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 $1$ (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: | $13$ | 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: |
$\frac{21763108131}{16953699937}a^{17}-\frac{58370006999}{16953699937}a^{16}+\frac{59187006340}{16953699937}a^{15}-\frac{14442945600}{16953699937}a^{14}-\frac{360711714461}{16953699937}a^{13}+\frac{676855893457}{16953699937}a^{12}-\frac{570130541237}{16953699937}a^{11}-\frac{65217494746}{16953699937}a^{10}+\frac{1958269709961}{16953699937}a^{9}-\frac{2795092055273}{16953699937}a^{8}-\frac{17560288051}{16953699937}a^{7}+\frac{2967416583262}{16953699937}a^{6}-\frac{1889571006018}{16953699937}a^{5}-\frac{545731009086}{16953699937}a^{4}+\frac{894459290979}{16953699937}a^{3}-\frac{183087835002}{16953699937}a^{2}-\frac{44359603763}{16953699937}a-\frac{6635736581}{16953699937}$, $\frac{29673612117}{16953699937}a^{17}-\frac{95248221527}{16953699937}a^{16}+\frac{112429098315}{16953699937}a^{15}-\frac{29518734839}{16953699937}a^{14}-\frac{523617784100}{16953699937}a^{13}+\frac{1208230087680}{16953699937}a^{12}-\frac{1110439027129}{16953699937}a^{11}-\frac{69650505939}{16953699937}a^{10}+\frac{3142428818648}{16953699937}a^{9}-\frac{5391745758800}{16953699937}a^{8}+\frac{1191115482078}{16953699937}a^{7}+\frac{5705079815132}{16953699937}a^{6}-\frac{5320421302705}{16953699937}a^{5}-\frac{473862800677}{16953699937}a^{4}+\frac{2674707384618}{16953699937}a^{3}-\frac{972855453405}{16953699937}a^{2}-\frac{156530747725}{16953699937}a+\frac{81826392329}{16953699937}$, $\frac{29673612117}{16953699937}a^{17}-\frac{95248221527}{16953699937}a^{16}+\frac{112429098315}{16953699937}a^{15}-\frac{29518734839}{16953699937}a^{14}-\frac{523617784100}{16953699937}a^{13}+\frac{1208230087680}{16953699937}a^{12}-\frac{1110439027129}{16953699937}a^{11}-\frac{69650505939}{16953699937}a^{10}+\frac{3142428818648}{16953699937}a^{9}-\frac{5391745758800}{16953699937}a^{8}+\frac{1191115482078}{16953699937}a^{7}+\frac{5705079815132}{16953699937}a^{6}-\frac{5320421302705}{16953699937}a^{5}-\frac{473862800677}{16953699937}a^{4}+\frac{2674707384618}{16953699937}a^{3}-\frac{972855453405}{16953699937}a^{2}-\frac{156530747725}{16953699937}a+\frac{98780092266}{16953699937}$, $\frac{5768035267}{16953699937}a^{17}-\frac{23709075801}{16953699937}a^{16}+\frac{29491824479}{16953699937}a^{15}-\frac{11882001542}{16953699937}a^{14}-\frac{93935554807}{16953699937}a^{13}+\frac{304036168711}{16953699937}a^{12}-\frac{264140526549}{16953699937}a^{11}+\frac{63565628290}{16953699937}a^{10}+\frac{562067058650}{16953699937}a^{9}-\frac{1291440070335}{16953699937}a^{8}+\frac{314496243708}{16953699937}a^{7}+\frac{1252443642448}{16953699937}a^{6}-\frac{862941604578}{16953699937}a^{5}-\frac{401893943044}{16953699937}a^{4}+\frac{413997327733}{16953699937}a^{3}+\frac{54360322625}{16953699937}a^{2}-\frac{79296888820}{16953699937}a-\frac{7829504610}{16953699937}$, $\frac{5954029154}{16953699937}a^{17}-\frac{28192967364}{16953699937}a^{16}+\frac{41554436059}{16953699937}a^{15}-\frac{17754463651}{16953699937}a^{14}-\frac{109679093628}{16953699937}a^{13}+\frac{392826431260}{16953699937}a^{12}-\frac{415406421027}{16953699937}a^{11}+\frac{80994606503}{16953699937}a^{10}+\frac{754998539717}{16953699937}a^{9}-\frac{1850866110259}{16953699937}a^{8}+\frac{926501621921}{16953699937}a^{7}+\frac{1717809948091}{16953699937}a^{6}-\frac{2135358058677}{16953699937}a^{5}+\frac{4849371762}{16953699937}a^{4}+\frac{1046167693023}{16953699937}a^{3}-\frac{370769943240}{16953699937}a^{2}-\frac{103179916560}{16953699937}a+\frac{35525490432}{16953699937}$, $\frac{12873346548}{16953699937}a^{17}-\frac{34295285417}{16953699937}a^{16}+\frac{27111490459}{16953699937}a^{15}+\frac{13269768246}{16953699937}a^{14}-\frac{238058201018}{16953699937}a^{13}+\frac{410299809560}{16953699937}a^{12}-\frac{219466618201}{16953699937}a^{11}-\frac{289388253553}{16953699937}a^{10}+\frac{1398477569142}{16953699937}a^{9}-\frac{1716084641393}{16953699937}a^{8}-\frac{611045177934}{16953699937}a^{7}+\frac{2744495431627}{16953699937}a^{6}-\frac{1288250147379}{16953699937}a^{5}-\frac{1059371069049}{16953699937}a^{4}+\frac{1080710939331}{16953699937}a^{3}-\frac{66919816907}{16953699937}a^{2}-\frac{153341869403}{16953699937}a+\frac{13762382301}{16953699937}$, $a^{17}-3a^{16}+3a^{15}-18a^{13}+37a^{12}-27a^{11}-12a^{10}+107a^{9}-158a^{8}-7a^{7}+207a^{6}-134a^{5}-64a^{4}+90a^{3}-10a^{2}-16a+2$, $\frac{40971713783}{16953699937}a^{17}-\frac{117304144785}{16953699937}a^{16}+\frac{120371097239}{16953699937}a^{15}-\frac{18034047089}{16953699937}a^{14}-\frac{709277563100}{16953699937}a^{13}+\frac{1419085963193}{16953699937}a^{12}-\frac{1139997446810}{16953699937}a^{11}-\frac{256816502223}{16953699937}a^{10}+\frac{4073696063415}{16953699937}a^{9}-\frac{6058414484608}{16953699937}a^{8}+\frac{110495315757}{16953699937}a^{7}+\frac{6920389476336}{16953699937}a^{6}-\frac{4935133513209}{16953699937}a^{5}-\frac{1233451906707}{16953699937}a^{4}+\frac{2667804340785}{16953699937}a^{3}-\frac{750259729245}{16953699937}a^{2}-\frac{177060593151}{16953699937}a+\frac{69112095905}{16953699937}$, $\frac{2728113540}{16953699937}a^{17}-\frac{16496115711}{16953699937}a^{16}+\frac{31140137798}{16953699937}a^{15}-\frac{22172729475}{16953699937}a^{14}-\frac{44707427637}{16953699937}a^{13}+\frac{237516639526}{16953699937}a^{12}-\frac{338764558931}{16953699937}a^{11}+\frac{167074732303}{16953699937}a^{10}+\frac{321654142106}{16953699937}a^{9}-\frac{1168632033457}{16953699937}a^{8}+\frac{1044766285309}{16953699937}a^{7}+\frac{682568195061}{16953699937}a^{6}-\frac{1559928163568}{16953699937}a^{5}+\frac{373475575724}{16953699937}a^{4}+\frac{615354824381}{16953699937}a^{3}-\frac{342675670865}{16953699937}a^{2}-\frac{45630994600}{16953699937}a+\frac{54295383054}{16953699937}$, $\frac{38647097172}{16953699937}a^{17}-\frac{121682240501}{16953699937}a^{16}+\frac{132238705049}{16953699937}a^{15}-\frac{21414735336}{16953699937}a^{14}-\frac{685767097116}{16953699937}a^{13}+\frac{1527669476962}{16953699937}a^{12}-\frac{1245498675414}{16953699937}a^{11}-\frac{225985362833}{16953699937}a^{10}+\frac{4109173889383}{16953699937}a^{9}-\frac{6666141942684}{16953699937}a^{8}+\frac{661324936311}{16953699937}a^{7}+\frac{7595285052868}{16953699937}a^{6}-\frac{5951418843344}{16953699937}a^{5}-\frac{1364243479923}{16953699937}a^{4}+\frac{3224967071079}{16953699937}a^{3}-\frac{839885236256}{16953699937}a^{2}-\frac{297389797947}{16953699937}a+\frac{49581678337}{16953699937}$, $\frac{12873346548}{16953699937}a^{17}-\frac{34295285417}{16953699937}a^{16}+\frac{27111490459}{16953699937}a^{15}+\frac{13269768246}{16953699937}a^{14}-\frac{238058201018}{16953699937}a^{13}+\frac{410299809560}{16953699937}a^{12}-\frac{219466618201}{16953699937}a^{11}-\frac{289388253553}{16953699937}a^{10}+\frac{1398477569142}{16953699937}a^{9}-\frac{1716084641393}{16953699937}a^{8}-\frac{611045177934}{16953699937}a^{7}+\frac{2744495431627}{16953699937}a^{6}-\frac{1288250147379}{16953699937}a^{5}-\frac{1059371069049}{16953699937}a^{4}+\frac{1080710939331}{16953699937}a^{3}-\frac{66919816907}{16953699937}a^{2}-\frac{170295569340}{16953699937}a+\frac{30716082238}{16953699937}$, $\frac{42038654315}{16953699937}a^{17}-\frac{127358170831}{16953699937}a^{16}+\frac{150274347999}{16953699937}a^{15}-\frac{45185769606}{16953699937}a^{14}-\frac{731093947285}{16953699937}a^{13}+\frac{1588313552199}{16953699937}a^{12}-\frac{1522173678692}{16953699937}a^{11}-\frac{50952765225}{16953699937}a^{10}+\frac{4293420767760}{16953699937}a^{9}-\frac{7058623333722}{16953699937}a^{8}+\frac{1655803886200}{16953699937}a^{7}+\frac{7184625399520}{16953699937}a^{6}-\frac{7098270923675}{16953699937}a^{5}-\frac{214187515932}{16953699937}a^{4}+\frac{3474640911885}{16953699937}a^{3}-\frac{1448815537338}{16953699937}a^{2}-\frac{216277828461}{16953699937}a+\frac{151176125224}{16953699937}$, $\frac{5954029154}{16953699937}a^{17}-\frac{28192967364}{16953699937}a^{16}+\frac{41554436059}{16953699937}a^{15}-\frac{17754463651}{16953699937}a^{14}-\frac{109679093628}{16953699937}a^{13}+\frac{392826431260}{16953699937}a^{12}-\frac{415406421027}{16953699937}a^{11}+\frac{80994606503}{16953699937}a^{10}+\frac{754998539717}{16953699937}a^{9}-\frac{1850866110259}{16953699937}a^{8}+\frac{926501621921}{16953699937}a^{7}+\frac{1717809948091}{16953699937}a^{6}-\frac{2135358058677}{16953699937}a^{5}+\frac{4849371762}{16953699937}a^{4}+\frac{1046167693023}{16953699937}a^{3}-\frac{370769943240}{16953699937}a^{2}-\frac{103179916560}{16953699937}a+\frac{52479190369}{16953699937}$
| 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: | \( 49273.3749081 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 49273.3749081 \cdot 1}{2\cdot\sqrt{66983486784702356675569}}\cr\approx \mathstrut & 0.151921020388 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 3*x^16 - 18*x^14 + 37*x^13 - 27*x^12 - 12*x^11 + 107*x^10 - 158*x^9 - 7*x^8 + 207*x^7 - 134*x^6 - 64*x^5 + 90*x^4 - 10*x^3 - 16*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 - 3*x^17 + 3*x^16 - 18*x^14 + 37*x^13 - 27*x^12 - 12*x^11 + 107*x^10 - 158*x^9 - 7*x^8 + 207*x^7 - 134*x^6 - 64*x^5 + 90*x^4 - 10*x^3 - 16*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 - 3*x^17 + 3*x^16 - 18*x^14 + 37*x^13 - 27*x^12 - 12*x^11 + 107*x^10 - 158*x^9 - 7*x^8 + 207*x^7 - 134*x^6 - 64*x^5 + 90*x^4 - 10*x^3 - 16*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 - 3*x^17 + 3*x^16 - 18*x^14 + 37*x^13 - 27*x^12 - 12*x^11 + 107*x^10 - 158*x^9 - 7*x^8 + 207*x^7 - 134*x^6 - 64*x^5 + 90*x^4 - 10*x^3 - 16*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
$S_4^3.A_4$ (as 18T838):
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 165888 |
| The 180 conjugacy class representatives for $S_4^3.A_4$ |
| Character table for $S_4^3.A_4$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.7.2668161671.1 |
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 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 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 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
|
\(97\)
| 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 97.4.0.1 | $x^{4} + 6 x^{2} + 80 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.6.0.1 | $x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(22679\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |