Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 47*x^16 - 47*x^15 + 899*x^14 + 1798*x^13 - 8043*x^12 - 26826*x^11 + 21861*x^10 + 185806*x^9 + 154879*x^8 - 491467*x^7 - 1167834*x^6 - 436976*x^5 + 1652420*x^4 + 2980937*x^3 + 2338635*x^2 + 935454*x + 155909)
gp: K = bnfinit(y^18 - 47*y^16 - 47*y^15 + 899*y^14 + 1798*y^13 - 8043*y^12 - 26826*y^11 + 21861*y^10 + 185806*y^9 + 154879*y^8 - 491467*y^7 - 1167834*y^6 - 436976*y^5 + 1652420*y^4 + 2980937*y^3 + 2338635*y^2 + 935454*y + 155909, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 47*x^16 - 47*x^15 + 899*x^14 + 1798*x^13 - 8043*x^12 - 26826*x^11 + 21861*x^10 + 185806*x^9 + 154879*x^8 - 491467*x^7 - 1167834*x^6 - 436976*x^5 + 1652420*x^4 + 2980937*x^3 + 2338635*x^2 + 935454*x + 155909);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 47*x^16 - 47*x^15 + 899*x^14 + 1798*x^13 - 8043*x^12 - 26826*x^11 + 21861*x^10 + 185806*x^9 + 154879*x^8 - 491467*x^7 - 1167834*x^6 - 436976*x^5 + 1652420*x^4 + 2980937*x^3 + 2338635*x^2 + 935454*x + 155909)
\( x^{18} - 47 x^{16} - 47 x^{15} + 899 x^{14} + 1798 x^{13} - 8043 x^{12} - 26826 x^{11} + 21861 x^{10} + \cdots + 155909 \)
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: | $[14, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(19779360901924343686926088019889763648\)
\(\medspace = 2^{6}\cdot 13^{2}\cdot 17^{9}\cdot 67^{2}\cdot 101^{7}\cdot 179^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(118.04\) | 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: | $2^{2/3}13^{2/3}17^{1/2}67^{2/3}101^{3/4}179^{2/3}\approx 604040.8065579127$ | ||
| Ramified primes: |
\(2\), \(13\), \(17\), \(67\), \(101\), \(179\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1717}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{6}-\frac{3}{8}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{7}-\frac{3}{8}a^{6}-\frac{1}{4}a^{5}+\frac{3}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{8}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{3}{8}a^{7}-\frac{3}{8}a^{6}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}-\frac{1}{8}a^{2}-\frac{1}{2}a+\frac{1}{8}$, $\frac{1}{32}a^{15}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{7}{32}a^{11}+\frac{1}{16}a^{10}-\frac{11}{32}a^{8}-\frac{1}{32}a^{7}+\frac{15}{32}a^{6}-\frac{13}{32}a^{5}+\frac{11}{32}a^{4}-\frac{5}{16}a^{2}+\frac{3}{32}a-\frac{9}{32}$, $\frac{1}{32}a^{16}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}+\frac{1}{32}a^{12}+\frac{1}{16}a^{11}-\frac{1}{4}a^{10}-\frac{3}{32}a^{9}-\frac{1}{32}a^{8}+\frac{15}{32}a^{7}+\frac{11}{32}a^{6}-\frac{13}{32}a^{5}-\frac{1}{2}a^{4}-\frac{1}{16}a^{3}+\frac{3}{32}a^{2}+\frac{7}{32}a-\frac{1}{4}$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{14}+\frac{1}{32}a^{13}-\frac{1}{16}a^{12}+\frac{3}{16}a^{11}-\frac{3}{32}a^{10}-\frac{1}{32}a^{9}-\frac{7}{32}a^{8}+\frac{5}{32}a^{7}+\frac{5}{32}a^{6}+\frac{7}{16}a^{5}-\frac{5}{32}a^{3}+\frac{3}{32}a^{2}+\frac{1}{16}a+\frac{7}{16}$
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: | $15$ | 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: |
$a+1$, $\frac{781}{16}a^{17}-\frac{221}{4}a^{16}-\frac{35703}{16}a^{15}+231a^{14}+\frac{697791}{16}a^{13}+\frac{307319}{8}a^{12}-\frac{6974519}{16}a^{11}-\frac{13057937}{16}a^{10}+\frac{31824415}{16}a^{9}+\frac{27269117}{4}a^{8}-\frac{1168635}{8}a^{7}-\frac{95235351}{4}a^{6}-\frac{481263805}{16}a^{5}+\frac{202292675}{16}a^{4}+\frac{1061110087}{16}a^{3}+\frac{1128625251}{16}a^{2}+\frac{551450133}{16}a+\frac{107865653}{16}$, $\frac{509}{16}a^{17}-\frac{149}{4}a^{16}-\frac{23223}{16}a^{15}+204a^{14}+\frac{453679}{16}a^{13}+\frac{192047}{8}a^{12}-\frac{4541911}{16}a^{11}-\frac{8336929}{16}a^{10}+\frac{20871055}{16}a^{9}+\frac{17531633}{4}a^{8}-\frac{1588731}{8}a^{7}-\frac{61570271}{4}a^{6}-\frac{306277853}{16}a^{5}+\frac{135483411}{16}a^{4}+\frac{682150279}{16}a^{3}+\frac{719520035}{16}a^{2}+\frac{349392069}{16}a+\frac{67951717}{16}$, $\frac{123}{4}a^{17}-\frac{73}{2}a^{16}-1402a^{15}+\frac{1753}{8}a^{14}+\frac{109549}{4}a^{13}+\frac{182195}{8}a^{12}-\frac{2195309}{8}a^{11}-\frac{1996523}{4}a^{10}+\frac{5061219}{4}a^{9}+\frac{33694549}{8}a^{8}-\frac{1923233}{8}a^{7}-\frac{29663341}{2}a^{6}-\frac{146377433}{8}a^{5}+\frac{33216001}{4}a^{4}+\frac{163842443}{4}a^{3}+\frac{344091163}{8}a^{2}+\frac{41627703}{2}a+\frac{8069167}{2}$, $3a^{17}-11a^{16}-116a^{15}+328a^{14}+2123a^{13}-3552a^{12}-22908a^{11}+12007a^{10}+145608a^{9}+63759a^{8}-480512a^{7}-621226a^{6}+486270a^{5}+1484549a^{4}+779422a^{3}-467727a^{2}-623636a-181894$, $\frac{207}{8}a^{17}-\frac{71}{2}a^{16}-\frac{9347}{8}a^{15}+\frac{3113}{8}a^{14}+\frac{182149}{8}a^{13}+\frac{121841}{8}a^{12}-\frac{459587}{2}a^{11}-\frac{3029059}{8}a^{10}+\frac{8747117}{8}a^{9}+\frac{26507157}{8}a^{8}-\frac{4675699}{8}a^{7}-11991414a^{6}-\frac{54673515}{4}a^{5}+\frac{62334177}{8}a^{4}+\frac{257825609}{8}a^{3}+\frac{129872197}{4}a^{2}+\frac{121764929}{8}a+\frac{22899405}{8}$, $\frac{2413}{32}a^{17}-\frac{3089}{32}a^{16}-\frac{109429}{32}a^{15}+\frac{3325}{4}a^{14}+\frac{2134089}{32}a^{13}+\frac{1608543}{32}a^{12}-\frac{21445265}{32}a^{11}-\frac{37290105}{32}a^{10}+\frac{50129907}{16}a^{9}+\frac{319896061}{32}a^{8}-\frac{34500537}{32}a^{7}-\frac{1139902085}{32}a^{6}-\frac{340424111}{8}a^{5}+\frac{679894617}{32}a^{4}+\frac{3113728001}{32}a^{3}+\frac{1609557507}{16}a^{2}+\frac{385186797}{8}a+\frac{295784059}{32}$, $\frac{2569}{16}a^{17}-\frac{3887}{16}a^{16}-\frac{114873}{16}a^{15}+\frac{26547}{8}a^{14}+\frac{2229665}{16}a^{13}+\frac{1244729}{16}a^{12}-\frac{22554659}{16}a^{11}-\frac{34785247}{16}a^{10}+\frac{54442813}{8}a^{9}+\frac{312632889}{16}a^{8}-\frac{75670321}{16}a^{7}-\frac{1148848605}{16}a^{6}-\frac{630352277}{8}a^{5}+\frac{788547445}{16}a^{4}+\frac{3053286111}{16}a^{3}+\frac{1516648663}{8}a^{2}+\frac{705416571}{8}a+\frac{264059729}{16}$, $\frac{1001}{32}a^{17}-\frac{295}{8}a^{16}-\frac{45655}{32}a^{15}+\frac{3385}{16}a^{14}+\frac{891875}{32}a^{13}+\frac{187121}{8}a^{12}-\frac{8932529}{32}a^{11}-\frac{16323021}{32}a^{10}+\frac{41115931}{32}a^{9}+\frac{68757815}{16}a^{8}-\frac{877991}{4}a^{7}-\frac{120896953}{8}a^{6}-\frac{599032719}{32}a^{5}+\frac{268347415}{32}a^{4}+\frac{1337520051}{32}a^{3}+\frac{1407702361}{32}a^{2}+\frac{682413725}{32}a+\frac{132505117}{32}$, $\frac{2527}{32}a^{17}-\frac{2941}{32}a^{16}-\frac{57667}{16}a^{15}+\frac{3857}{8}a^{14}+\frac{2253309}{32}a^{13}+\frac{1921905}{32}a^{12}-\frac{11275891}{16}a^{11}-\frac{41547889}{32}a^{10}+\frac{6468493}{2}a^{9}+\frac{174514959}{16}a^{8}-\frac{7128369}{16}a^{7}-\frac{612254703}{16}a^{6}-\frac{1527306951}{32}a^{5}+\frac{334649323}{16}a^{4}+\frac{3395241383}{32}a^{3}+\frac{1793396477}{16}a^{2}+\frac{1743691497}{32}a+\frac{42433923}{4}$, $\frac{637}{16}a^{17}-\frac{1963}{32}a^{16}-\frac{56883}{32}a^{15}+\frac{13925}{16}a^{14}+\frac{551819}{16}a^{13}+\frac{588007}{32}a^{12}-\frac{11175869}{32}a^{11}-\frac{4235461}{8}a^{10}+\frac{54197727}{32}a^{9}+\frac{76652911}{16}a^{8}-\frac{5035357}{4}a^{7}-\frac{141484595}{8}a^{6}-\frac{306285991}{16}a^{5}+\frac{396368175}{32}a^{4}+\frac{374357063}{8}a^{3}+\frac{1477566067}{32}a^{2}+\frac{341833503}{16}a+\frac{127338519}{32}$, $\frac{3503}{16}a^{17}-\frac{995}{4}a^{16}-\frac{80057}{8}a^{15}+\frac{8631}{8}a^{14}+\frac{3129375}{16}a^{13}+\frac{1371623}{8}a^{12}-\frac{3910933}{2}a^{11}-\frac{58425869}{16}a^{10}+\frac{142919237}{16}a^{9}+\frac{488479189}{16}a^{8}-\frac{12218487}{16}a^{7}-\frac{1707395747}{16}a^{6}-\frac{537893635}{4}a^{5}+\frac{456115215}{8}a^{4}+\frac{4751425925}{16}a^{3}+\frac{5045997241}{16}a^{2}+153903094a+\frac{120267825}{4}$, $a^{17}-\frac{21}{8}a^{16}-\frac{659}{16}a^{15}+\frac{513}{8}a^{14}+\frac{6197}{8}a^{13}-\frac{2495}{8}a^{12}-\frac{128781}{16}a^{11}-\frac{41781}{8}a^{10}+\frac{353661}{8}a^{9}+\frac{1183053}{16}a^{8}-\frac{1407533}{16}a^{7}-\frac{5316427}{16}a^{6}-\frac{2984633}{16}a^{5}+\frac{6322091}{16}a^{4}+\frac{1506215}{2}a^{3}+\frac{4344595}{8}a^{2}+\frac{2872369}{16}a+\frac{329599}{16}$, $\frac{731}{32}a^{17}-\frac{617}{32}a^{16}-\frac{16985}{16}a^{15}-\frac{1335}{8}a^{14}+\frac{667301}{32}a^{13}+\frac{742313}{32}a^{12}-\frac{3306519}{16}a^{11}-\frac{13977789}{32}a^{10}+\frac{7226021}{8}a^{9}+\frac{56014005}{16}a^{8}+\frac{6151331}{16}a^{7}-\frac{189550609}{16}a^{6}-\frac{519407259}{32}a^{5}+\frac{81791079}{16}a^{4}+\frac{1087069335}{32}a^{3}+\frac{600602877}{16}a^{2}+\frac{601722261}{32}a+\frac{7509227}{2}$, $\frac{45591}{16}a^{17}-\frac{102397}{32}a^{16}-\frac{4170671}{32}a^{15}+\frac{199187}{16}a^{14}+\frac{40764895}{16}a^{13}+\frac{72380413}{32}a^{12}-\frac{814746093}{32}a^{11}-\frac{191381213}{4}a^{10}+\frac{3713600137}{32}a^{9}+\frac{1596534611}{4}a^{8}-\frac{113084295}{16}a^{7}-\frac{22283262633}{16}a^{6}-\frac{7053228505}{4}a^{5}+\frac{23557465955}{32}a^{4}+\frac{31057417097}{8}a^{3}+\frac{132249780401}{32}a^{2}+\frac{16162515259}{8}a+\frac{12650140819}{32}$
| 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: | \( 5242948282790 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 5242948282790 \cdot 1}{2\cdot\sqrt{19779360901924343686926088019889763648}}\cr\approx \mathstrut & 0.381258141684155 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 47*x^16 - 47*x^15 + 899*x^14 + 1798*x^13 - 8043*x^12 - 26826*x^11 + 21861*x^10 + 185806*x^9 + 154879*x^8 - 491467*x^7 - 1167834*x^6 - 436976*x^5 + 1652420*x^4 + 2980937*x^3 + 2338635*x^2 + 935454*x + 155909)
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 - 47*x^16 - 47*x^15 + 899*x^14 + 1798*x^13 - 8043*x^12 - 26826*x^11 + 21861*x^10 + 185806*x^9 + 154879*x^8 - 491467*x^7 - 1167834*x^6 - 436976*x^5 + 1652420*x^4 + 2980937*x^3 + 2338635*x^2 + 935454*x + 155909, 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 - 47*x^16 - 47*x^15 + 899*x^14 + 1798*x^13 - 8043*x^12 - 26826*x^11 + 21861*x^10 + 185806*x^9 + 154879*x^8 - 491467*x^7 - 1167834*x^6 - 436976*x^5 + 1652420*x^4 + 2980937*x^3 + 2338635*x^2 + 935454*x + 155909);
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 - 47*x^16 - 47*x^15 + 899*x^14 + 1798*x^13 - 8043*x^12 - 26826*x^11 + 21861*x^10 + 185806*x^9 + 154879*x^8 - 491467*x^7 - 1167834*x^6 - 436976*x^5 + 1652420*x^4 + 2980937*x^3 + 2338635*x^2 + 935454*x + 155909);
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
$C_3^6.A_4^2:D_4$ (as 18T921):
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 839808 |
| The 98 conjugacy class representatives for $C_3^6.A_4^2:D_4$ are not computed |
| Character table for $C_3^6.A_4^2:D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), 6.6.200470052.1 |
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 24 sibling: | 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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | $18$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | R | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(17\)
| 17.18.9.1 | $x^{18} + 1377 x^{17} + 842877 x^{16} + 301039740 x^{15} + 69145935966 x^{14} + 10594681229538 x^{13} + 1083400274206194 x^{12} + 71374592916053006 x^{11} + 2757704886031296688 x^{10} + 48363988493127014880 x^{9} + 46880983064641450752 x^{8} + 20627257837536901014 x^{7} + 5322819889168442136 x^{6} + 892489846394695920 x^{5} + 600661719561810931 x^{4} + 19503284078495824915 x^{3} + 346236970687224298200 x^{2} + 476441885309573581545 x + 683564390300988440775$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.3.2.3 | $x^{3} + 134$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 67.4.0.1 | $x^{4} + 8 x^{2} + 54 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 67.4.0.1 | $x^{4} + 8 x^{2} + 54 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(101\)
| $\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.4.3.1 | $x^{4} + 404$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 101.6.3.1 | $x^{6} + 24543 x^{5} + 200786592 x^{4} + 547549249797 x^{3} + 20884233801 x^{2} + 1662760800612 x + 54226912599047$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(179\)
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.3.2.1 | $x^{3} + 179$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 179.3.0.1 | $x^{3} + 4 x + 177$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |