Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 5*x^18 + 2*x^17 + 25*x^16 - 2*x^15 - 55*x^14 + 72*x^13 - 35*x^12 - 70*x^11 + 139*x^10 - 70*x^9 - 35*x^8 + 72*x^7 - 55*x^6 - 2*x^5 + 25*x^4 + 2*x^3 - 5*x^2 - 2*x + 1)
gp: K = bnfinit(y^20 - 2*y^19 - 5*y^18 + 2*y^17 + 25*y^16 - 2*y^15 - 55*y^14 + 72*y^13 - 35*y^12 - 70*y^11 + 139*y^10 - 70*y^9 - 35*y^8 + 72*y^7 - 55*y^6 - 2*y^5 + 25*y^4 + 2*y^3 - 5*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 - 5*x^18 + 2*x^17 + 25*x^16 - 2*x^15 - 55*x^14 + 72*x^13 - 35*x^12 - 70*x^11 + 139*x^10 - 70*x^9 - 35*x^8 + 72*x^7 - 55*x^6 - 2*x^5 + 25*x^4 + 2*x^3 - 5*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 - 5*x^18 + 2*x^17 + 25*x^16 - 2*x^15 - 55*x^14 + 72*x^13 - 35*x^12 - 70*x^11 + 139*x^10 - 70*x^9 - 35*x^8 + 72*x^7 - 55*x^6 - 2*x^5 + 25*x^4 + 2*x^3 - 5*x^2 - 2*x + 1)
\( x^{20} - 2 x^{19} - 5 x^{18} + 2 x^{17} + 25 x^{16} - 2 x^{15} - 55 x^{14} + 72 x^{13} - 35 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(100230360358533121932197761\)
\(\medspace = 43^{2}\cdot 61^{4}\cdot 397^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.95\) | 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: | $43^{1/2}61^{1/2}397^{1/2}\approx 1020.4562704986432$ | ||
| Ramified primes: |
\(43\), \(61\), \(397\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{6}$, $\frac{1}{1216702}a^{18}-\frac{72933}{608351}a^{17}+\frac{65172}{608351}a^{16}-\frac{82948}{608351}a^{15}-\frac{126951}{608351}a^{14}+\frac{67522}{608351}a^{13}-\frac{3570}{608351}a^{12}-\frac{81462}{608351}a^{11}+\frac{129977}{1216702}a^{10}+\frac{456641}{1216702}a^{9}+\frac{129977}{1216702}a^{8}-\frac{81462}{608351}a^{7}+\frac{601211}{1216702}a^{6}-\frac{473307}{1216702}a^{5}+\frac{354449}{1216702}a^{4}+\frac{442455}{1216702}a^{3}-\frac{478007}{1216702}a^{2}+\frac{462485}{1216702}a+\frac{1}{1216702}$, $\frac{1}{1216702}a^{19}-\frac{145869}{608351}a^{17}-\frac{201795}{1216702}a^{16}+\frac{73120}{608351}a^{15}+\frac{266441}{1216702}a^{14}-\frac{42208}{608351}a^{13}-\frac{74626}{608351}a^{12}+\frac{144804}{608351}a^{11}-\frac{185543}{1216702}a^{10}+\frac{175093}{1216702}a^{9}+\frac{205797}{608351}a^{8}+\frac{152491}{1216702}a^{7}-\frac{459635}{1216702}a^{6}-\frac{165589}{608351}a^{5}-\frac{34499}{1216702}a^{4}+\frac{265243}{608351}a^{3}-\frac{181765}{1216702}a^{2}-\frac{422081}{1216702}a-\frac{462485}{1216702}$
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: |
$\frac{717658}{608351}a^{19}-\frac{2047767}{1216702}a^{18}-\frac{8377335}{1216702}a^{17}-\frac{1996665}{1216702}a^{16}+\frac{17553038}{608351}a^{15}+\frac{8839816}{608351}a^{14}-\frac{34734090}{608351}a^{13}+\frac{61112703}{1216702}a^{12}-\frac{6786846}{608351}a^{11}-\frac{106271145}{1216702}a^{10}+\frac{131258047}{1216702}a^{9}-\frac{14901361}{1216702}a^{8}-\frac{31025218}{608351}a^{7}+\frac{30344514}{608351}a^{6}-\frac{16839207}{608351}a^{5}-\frac{29350011}{1216702}a^{4}+\frac{19129563}{1216702}a^{3}+\frac{18615191}{1216702}a^{2}+\frac{552084}{608351}a-\frac{2240761}{1216702}$, $\frac{6334393}{1216702}a^{19}-\frac{10170463}{1216702}a^{18}-\frac{17728071}{608351}a^{17}-\frac{869144}{608351}a^{16}+\frac{156576003}{1216702}a^{15}+\frac{49133575}{1216702}a^{14}-\frac{161801721}{608351}a^{13}+\frac{329012779}{1216702}a^{12}-\frac{102220353}{1216702}a^{11}-\frac{470268259}{1216702}a^{10}+\frac{685351767}{1216702}a^{9}-\frac{90844236}{608351}a^{8}-\frac{136533649}{608351}a^{7}+\frac{338464557}{1216702}a^{6}-\frac{218518401}{1216702}a^{5}-\frac{46058160}{608351}a^{4}+\frac{58653122}{608351}a^{3}+\frac{55045585}{1216702}a^{2}-\frac{2595079}{608351}a-\frac{6329297}{608351}$, $a^{19}-2a^{18}-5a^{17}+2a^{16}+25a^{15}-2a^{14}-55a^{13}+72a^{12}-35a^{11}-70a^{10}+139a^{9}-70a^{8}-35a^{7}+72a^{6}-55a^{5}-2a^{4}+25a^{3}+2a^{2}-5a-2$, $\frac{3894933}{1216702}a^{19}-\frac{6419477}{1216702}a^{18}-\frac{21518137}{1216702}a^{17}-\frac{424019}{1216702}a^{16}+\frac{48313200}{608351}a^{15}+\frac{27389839}{1216702}a^{14}-\frac{99732478}{608351}a^{13}+\frac{102691218}{608351}a^{12}-\frac{74180475}{1216702}a^{11}-\frac{137944761}{608351}a^{10}+\frac{209638136}{608351}a^{9}-\frac{122586985}{1216702}a^{8}-\frac{151359929}{1216702}a^{7}+\frac{195850191}{1216702}a^{6}-\frac{133033763}{1216702}a^{5}-\frac{24750903}{608351}a^{4}+\frac{67647407}{1216702}a^{3}+\frac{18022220}{608351}a^{2}-\frac{938255}{608351}a-\frac{8338101}{1216702}$, $\frac{886043}{608351}a^{19}-\frac{1288411}{608351}a^{18}-\frac{10134921}{1216702}a^{17}-\frac{1171311}{608351}a^{16}+\frac{42688691}{1216702}a^{15}+\frac{20072863}{1216702}a^{14}-\frac{83739621}{1216702}a^{13}+\frac{79753207}{1216702}a^{12}-\frac{22524785}{1216702}a^{11}-\frac{62153986}{608351}a^{10}+\frac{81430523}{608351}a^{9}-\frac{13983576}{608351}a^{8}-\frac{35505482}{608351}a^{7}+\frac{76690389}{1216702}a^{6}-\frac{22491769}{608351}a^{5}-\frac{33585947}{1216702}a^{4}+\frac{27504739}{1216702}a^{3}+\frac{17768095}{1216702}a^{2}+\frac{517579}{1216702}a-\frac{3211895}{1216702}$, $\frac{6334393}{1216702}a^{19}-\frac{10170463}{1216702}a^{18}-\frac{17728071}{608351}a^{17}-\frac{869144}{608351}a^{16}+\frac{156576003}{1216702}a^{15}+\frac{49133575}{1216702}a^{14}-\frac{161801721}{608351}a^{13}+\frac{329012779}{1216702}a^{12}-\frac{102220353}{1216702}a^{11}-\frac{470268259}{1216702}a^{10}+\frac{685351767}{1216702}a^{9}-\frac{90844236}{608351}a^{8}-\frac{136533649}{608351}a^{7}+\frac{338464557}{1216702}a^{6}-\frac{218518401}{1216702}a^{5}-\frac{46058160}{608351}a^{4}+\frac{58653122}{608351}a^{3}+\frac{55045585}{1216702}a^{2}-\frac{2595079}{608351}a-\frac{6937648}{608351}$, $\frac{4349100}{608351}a^{19}-\frac{6688159}{608351}a^{18}-\frac{24928901}{608351}a^{17}-\frac{5282205}{1216702}a^{16}+\frac{215867701}{1216702}a^{15}+\frac{82211995}{1216702}a^{14}-\frac{444729069}{1216702}a^{13}+\frac{420761873}{1216702}a^{12}-\frac{50777997}{608351}a^{11}-\frac{668044129}{1216702}a^{10}+\frac{909827673}{1216702}a^{9}-\frac{182412193}{1216702}a^{8}-\frac{407727149}{1216702}a^{7}+\frac{451904279}{1216702}a^{6}-\frac{136289617}{608351}a^{5}-\frac{73213746}{608351}a^{4}+\frac{77719202}{608351}a^{3}+\frac{44157655}{608351}a^{2}-\frac{1707341}{608351}a-\frac{20405771}{1216702}$, $\frac{10283345}{1216702}a^{19}-\frac{8072797}{608351}a^{18}-\frac{29147371}{608351}a^{17}-\frac{2270089}{608351}a^{16}+\frac{127301643}{608351}a^{15}+\frac{88770415}{1216702}a^{14}-\frac{262817662}{608351}a^{13}+\frac{516096725}{1216702}a^{12}-\frac{142603289}{1216702}a^{11}-\frac{389766086}{608351}a^{10}+\frac{549555650}{608351}a^{9}-\frac{260240741}{1216702}a^{8}-\frac{230115750}{608351}a^{7}+\frac{272091519}{608351}a^{6}-\frac{171409069}{608351}a^{5}-\frac{79651381}{608351}a^{4}+\frac{185394737}{1216702}a^{3}+\frac{48790780}{608351}a^{2}-\frac{6506121}{1216702}a-\frac{21595749}{1216702}$, $\frac{636357}{1216702}a^{19}-\frac{912615}{1216702}a^{18}-\frac{1888417}{608351}a^{17}-\frac{237553}{608351}a^{16}+\frac{15468655}{1216702}a^{15}+\frac{6566481}{1216702}a^{14}-\frac{16368919}{608351}a^{13}+\frac{32294351}{1216702}a^{12}-\frac{1194469}{608351}a^{11}-\frac{59649885}{1216702}a^{10}+\frac{78257039}{1216702}a^{9}-\frac{11529}{787}a^{8}-\frac{20146945}{608351}a^{7}+\frac{52511075}{1216702}a^{6}-\frac{32078699}{1216702}a^{5}-\frac{5553292}{608351}a^{4}+\frac{9595699}{608351}a^{3}-\frac{2239601}{1216702}a^{2}+\frac{404958}{608351}a-\frac{258033}{1216702}$, $\frac{7298495}{1216702}a^{19}-\frac{11318577}{1216702}a^{18}-\frac{20722125}{608351}a^{17}-\frac{4168655}{1216702}a^{16}+\frac{179881019}{1216702}a^{15}+\frac{32837817}{608351}a^{14}-\frac{184706761}{608351}a^{13}+\frac{361586645}{1216702}a^{12}-\frac{48370843}{608351}a^{11}-\frac{548466207}{1216702}a^{10}+\frac{765246725}{1216702}a^{9}-\frac{86054866}{608351}a^{8}-\frac{162133454}{608351}a^{7}+\frac{375590853}{1216702}a^{6}-\frac{115824069}{608351}a^{5}-\frac{58607018}{608351}a^{4}+\frac{127264935}{1216702}a^{3}+\frac{70943015}{1216702}a^{2}-\frac{2812889}{608351}a-\frac{15996695}{1216702}$, $\frac{3426061}{608351}a^{19}-\frac{9960931}{1216702}a^{18}-\frac{20100072}{608351}a^{17}-\frac{7248803}{1216702}a^{16}+\frac{84881873}{608351}a^{15}+\frac{78070397}{1216702}a^{14}-\frac{346370157}{1216702}a^{13}+\frac{152605340}{608351}a^{12}-\frac{23510650}{608351}a^{11}-\frac{541697673}{1216702}a^{10}+\frac{678457925}{1216702}a^{9}-\frac{78232211}{1216702}a^{8}-\frac{176337284}{608351}a^{7}+\frac{344283047}{1216702}a^{6}-\frac{91291757}{608351}a^{5}-\frac{144680831}{1216702}a^{4}+\frac{61727826}{608351}a^{3}+\frac{38231619}{608351}a^{2}-\frac{608565}{1216702}a-\frac{15908941}{1216702}$, $\frac{1334573}{608351}a^{19}-\frac{4062661}{1216702}a^{18}-\frac{15144409}{1216702}a^{17}-\frac{2257773}{1216702}a^{16}+\frac{65272545}{1216702}a^{15}+\frac{13045971}{608351}a^{14}-\frac{65857708}{608351}a^{13}+\frac{63692160}{608351}a^{12}-\frac{17844508}{608351}a^{11}-\frac{95212780}{608351}a^{10}+\frac{130442176}{608351}a^{9}-\frac{28152841}{608351}a^{8}-\frac{105763161}{1216702}a^{7}+\frac{120408149}{1216702}a^{6}-\frac{75566855}{1216702}a^{5}-\frac{42172283}{1216702}a^{4}+\frac{19893393}{608351}a^{3}+\frac{13721612}{608351}a^{2}-\frac{331416}{608351}a-\frac{2180182}{608351}$, $\frac{2190710}{608351}a^{19}-\frac{3343628}{608351}a^{18}-\frac{12691047}{608351}a^{17}-\frac{1269859}{608351}a^{16}+\frac{109502867}{1216702}a^{15}+\frac{21129030}{608351}a^{14}-\frac{113870111}{608351}a^{13}+\frac{105359996}{608351}a^{12}-\frac{20146693}{608351}a^{11}-\frac{176730793}{608351}a^{10}+\frac{233393142}{608351}a^{9}-\frac{39435325}{608351}a^{8}-\frac{116809355}{608351}a^{7}+\frac{123856455}{608351}a^{6}-\frac{67876100}{608351}a^{5}-\frac{89464699}{1216702}a^{4}+\frac{46464761}{608351}a^{3}+\frac{20802798}{608351}a^{2}-\frac{3137038}{608351}a-\frac{3593591}{608351}$
| 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: | \( 253607.024173 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 253607.024173 \cdot 1}{2\cdot\sqrt{100230360358533121932197761}}\cr\approx \mathstrut & 0.199503634510 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 5*x^18 + 2*x^17 + 25*x^16 - 2*x^15 - 55*x^14 + 72*x^13 - 35*x^12 - 70*x^11 + 139*x^10 - 70*x^9 - 35*x^8 + 72*x^7 - 55*x^6 - 2*x^5 + 25*x^4 + 2*x^3 - 5*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^20 - 2*x^19 - 5*x^18 + 2*x^17 + 25*x^16 - 2*x^15 - 55*x^14 + 72*x^13 - 35*x^12 - 70*x^11 + 139*x^10 - 70*x^9 - 35*x^8 + 72*x^7 - 55*x^6 - 2*x^5 + 25*x^4 + 2*x^3 - 5*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^20 - 2*x^19 - 5*x^18 + 2*x^17 + 25*x^16 - 2*x^15 - 55*x^14 + 72*x^13 - 35*x^12 - 70*x^11 + 139*x^10 - 70*x^9 - 35*x^8 + 72*x^7 - 55*x^6 - 2*x^5 + 25*x^4 + 2*x^3 - 5*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^20 - 2*x^19 - 5*x^18 + 2*x^17 + 25*x^16 - 2*x^15 - 55*x^14 + 72*x^13 - 35*x^12 - 70*x^11 + 139*x^10 - 70*x^9 - 35*x^8 + 72*x^7 - 55*x^6 - 2*x^5 + 25*x^4 + 2*x^3 - 5*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
$C_2^{10}.S_5$ (as 20T799):
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 non-solvable group of order 122880 |
| The 252 conjugacy class representatives for $C_2^{10}.S_5$ |
| Character table for $C_2^{10}.S_5$ |
Intermediate fields
| 5.5.24217.1, 10.8.10011511392319.1, 10.4.25217912827.1, 10.6.232825846333.1 |
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 20 siblings: | data not computed |
| Degree 40 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(43\)
| 43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(61\)
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(397\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |