Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 14*x^18 - 33*x^17 + 85*x^16 - 156*x^15 + 281*x^14 - 417*x^13 + 590*x^12 - 734*x^11 + 869*x^10 - 875*x^9 + 908*x^8 - 679*x^7 + 668*x^6 - 321*x^5 + 303*x^4 - 77*x^3 + 42*x^2 - 4*x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 14*y^18 - 33*y^17 + 85*y^16 - 156*y^15 + 281*y^14 - 417*y^13 + 590*y^12 - 734*y^11 + 869*y^10 - 875*y^9 + 908*y^8 - 679*y^7 + 668*y^6 - 321*y^5 + 303*y^4 - 77*y^3 + 42*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 14*x^18 - 33*x^17 + 85*x^16 - 156*x^15 + 281*x^14 - 417*x^13 + 590*x^12 - 734*x^11 + 869*x^10 - 875*x^9 + 908*x^8 - 679*x^7 + 668*x^6 - 321*x^5 + 303*x^4 - 77*x^3 + 42*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 14*x^18 - 33*x^17 + 85*x^16 - 156*x^15 + 281*x^14 - 417*x^13 + 590*x^12 - 734*x^11 + 869*x^10 - 875*x^9 + 908*x^8 - 679*x^7 + 668*x^6 - 321*x^5 + 303*x^4 - 77*x^3 + 42*x^2 - 4*x + 1)
\( x^{20} - 3 x^{19} + 14 x^{18} - 33 x^{17} + 85 x^{16} - 156 x^{15} + 281 x^{14} - 417 x^{13} + 590 x^{12} + \cdots + 1 \)
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: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6802209109663753569286129\)
\(\medspace = 11^{16}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.44\) | 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: | $11^{4/5}23^{1/2}\approx 32.65713384043754$ | ||
| Ramified primes: |
\(11\), \(23\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{2232364087012}a^{19}-\frac{270591666149}{2232364087012}a^{18}-\frac{151793213265}{2232364087012}a^{17}-\frac{113404366523}{1116182043506}a^{16}+\frac{51411873400}{558091021753}a^{15}+\frac{151862348213}{2232364087012}a^{14}+\frac{542940291335}{2232364087012}a^{13}-\frac{87751579339}{2232364087012}a^{12}-\frac{8667138173}{1116182043506}a^{11}+\frac{9173191654}{558091021753}a^{10}-\frac{1028877853947}{2232364087012}a^{9}+\frac{360309042461}{1116182043506}a^{8}-\frac{632561197}{51915443884}a^{7}-\frac{946279076613}{2232364087012}a^{6}-\frac{473150293719}{2232364087012}a^{5}+\frac{900151682235}{2232364087012}a^{4}-\frac{392659928909}{2232364087012}a^{3}-\frac{54998655469}{558091021753}a^{2}+\frac{113225624400}{558091021753}a+\frac{502690245821}{1116182043506}$
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
$C_{2}$, which has order $2$
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: | $9$ | 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: |
$\frac{150533044601}{558091021753}a^{19}-\frac{407362055538}{558091021753}a^{18}+\frac{3924624100485}{1116182043506}a^{17}-\frac{8643401071791}{1116182043506}a^{16}+\frac{44798951553933}{2232364087012}a^{15}-\frac{38931039131165}{1116182043506}a^{14}+\frac{138909305035985}{2232364087012}a^{13}-\frac{197197200157891}{2232364087012}a^{12}+\frac{274107592289165}{2232364087012}a^{11}-\frac{163911898030829}{1116182043506}a^{10}+\frac{380003040752453}{2232364087012}a^{9}-\frac{179205422819619}{1116182043506}a^{8}+\frac{2186329596047}{12978860971}a^{7}-\frac{118627675893473}{1116182043506}a^{6}+\frac{270704948057963}{2232364087012}a^{5}-\frac{75477060472445}{2232364087012}a^{4}+\frac{60157834729907}{1116182043506}a^{3}-\frac{1115549374481}{2232364087012}a^{2}+\frac{5960610812087}{1116182043506}a-\frac{1578467688507}{2232364087012}$, $\frac{308576202581}{2232364087012}a^{19}-\frac{373383760777}{1116182043506}a^{18}+\frac{1879444905993}{1116182043506}a^{17}-\frac{7632825944615}{2232364087012}a^{16}+\frac{10016892608203}{1116182043506}a^{15}-\frac{32381003889593}{2232364087012}a^{14}+\frac{28618028119599}{1116182043506}a^{13}-\frac{37972470513423}{1116182043506}a^{12}+\frac{103165433913763}{2232364087012}a^{11}-\frac{57946809483181}{1116182043506}a^{10}+\frac{131587722479105}{2232364087012}a^{9}-\frac{111371916649365}{2232364087012}a^{8}+\frac{2860913159977}{51915443884}a^{7}-\frac{12909522960382}{558091021753}a^{6}+\frac{22424481235004}{558091021753}a^{5}+\frac{3734479940601}{1116182043506}a^{4}+\frac{11058369624822}{558091021753}a^{3}+\frac{17680353575405}{2232364087012}a^{2}+\frac{1560353205027}{1116182043506}a+\frac{797968092947}{1116182043506}$, $\frac{61300938463}{558091021753}a^{19}-\frac{601711179187}{2232364087012}a^{18}+\frac{3018478422139}{2232364087012}a^{17}-\frac{6144337697133}{2232364087012}a^{16}+\frac{16066842415073}{2232364087012}a^{15}-\frac{6463960022509}{558091021753}a^{14}+\frac{11302949917433}{558091021753}a^{13}-\frac{29558917354827}{1116182043506}a^{12}+\frac{19692994208708}{558091021753}a^{11}-\frac{21718253945382}{558091021753}a^{10}+\frac{96946072365691}{2232364087012}a^{9}-\frac{80177743296649}{2232364087012}a^{8}+\frac{519485330003}{12978860971}a^{7}-\frac{34399119181227}{2232364087012}a^{6}+\frac{16971188407641}{558091021753}a^{5}+\frac{1308967400144}{558091021753}a^{4}+\frac{37178537299871}{2232364087012}a^{3}+\frac{1828328535638}{558091021753}a^{2}+\frac{1886893259049}{1116182043506}a-\frac{824786303255}{2232364087012}$, $\frac{150533044601}{558091021753}a^{19}-\frac{407362055538}{558091021753}a^{18}+\frac{3924624100485}{1116182043506}a^{17}-\frac{8643401071791}{1116182043506}a^{16}+\frac{44798951553933}{2232364087012}a^{15}-\frac{38931039131165}{1116182043506}a^{14}+\frac{138909305035985}{2232364087012}a^{13}-\frac{197197200157891}{2232364087012}a^{12}+\frac{274107592289165}{2232364087012}a^{11}-\frac{163911898030829}{1116182043506}a^{10}+\frac{380003040752453}{2232364087012}a^{9}-\frac{179205422819619}{1116182043506}a^{8}+\frac{2186329596047}{12978860971}a^{7}-\frac{118627675893473}{1116182043506}a^{6}+\frac{270704948057963}{2232364087012}a^{5}-\frac{75477060472445}{2232364087012}a^{4}+\frac{60157834729907}{1116182043506}a^{3}-\frac{1115549374481}{2232364087012}a^{2}+\frac{5960610812087}{1116182043506}a+\frac{653896398505}{2232364087012}$, $\frac{18461088767}{2232364087012}a^{19}+\frac{141025247809}{1116182043506}a^{18}-\frac{637656410033}{2232364087012}a^{17}+\frac{1871952652965}{1116182043506}a^{16}-\frac{7984091900337}{2232364087012}a^{15}+\frac{22086123925521}{2232364087012}a^{14}-\frac{9636021487404}{558091021753}a^{13}+\frac{71572719864397}{2232364087012}a^{12}-\frac{52073373796873}{1116182043506}a^{11}+\frac{150458745125267}{2232364087012}a^{10}-\frac{186100007862163}{2232364087012}a^{9}+\frac{224610195899781}{2232364087012}a^{8}-\frac{1278671866284}{12978860971}a^{7}+\frac{59996395012757}{558091021753}a^{6}-\frac{162239572433181}{2232364087012}a^{5}+\frac{183426572508781}{2232364087012}a^{4}-\frac{63884546050923}{2232364087012}a^{3}+\frac{21142391233239}{558091021753}a^{2}-\frac{7733995785163}{2232364087012}a+\frac{2252441595874}{558091021753}$, $\frac{102099202399}{1116182043506}a^{19}-\frac{200367028729}{558091021753}a^{18}+\frac{3382055584637}{2232364087012}a^{17}-\frac{9306914222159}{2232364087012}a^{16}+\frac{23280078892223}{2232364087012}a^{15}-\frac{23753763382957}{1116182043506}a^{14}+\frac{43082174184467}{1116182043506}a^{13}-\frac{137978289707013}{2232364087012}a^{12}+\frac{199525733970277}{2232364087012}a^{11}-\frac{261960583659583}{2232364087012}a^{10}+\frac{78892496505866}{558091021753}a^{9}-\frac{169590678151585}{1116182043506}a^{8}+\frac{7958264176161}{51915443884}a^{7}-\frac{149498922687605}{1116182043506}a^{6}+\frac{246967341194407}{2232364087012}a^{5}-\frac{173666530990921}{2232364087012}a^{4}+\frac{101640326212061}{2232364087012}a^{3}-\frac{56384340034515}{2232364087012}a^{2}+\frac{13949032220553}{2232364087012}a-\frac{626778957683}{1116182043506}$, $\frac{50173726589}{1116182043506}a^{19}+\frac{25558497193}{558091021753}a^{18}+\frac{86925585465}{1116182043506}a^{17}+\frac{2348629678247}{2232364087012}a^{16}-\frac{2447285184319}{1116182043506}a^{15}+\frac{18559702906275}{2232364087012}a^{14}-\frac{34569318451979}{2232364087012}a^{13}+\frac{69951992026873}{2232364087012}a^{12}-\frac{52901914185789}{1116182043506}a^{11}+\frac{156949885182351}{2232364087012}a^{10}-\frac{49275892668534}{558091021753}a^{9}+\frac{121911504894795}{1116182043506}a^{8}-\frac{2749698544875}{25957721942}a^{7}+\frac{264454846336299}{2232364087012}a^{6}-\frac{171630699419409}{2232364087012}a^{5}+\frac{98560717521275}{1116182043506}a^{4}-\frac{69370302642253}{2232364087012}a^{3}+\frac{20464683103420}{558091021753}a^{2}-\frac{12938089864531}{2232364087012}a+\frac{1130336583239}{1116182043506}$, $\frac{63974657171}{1116182043506}a^{19}-\frac{51349129845}{558091021753}a^{18}+\frac{324703227816}{558091021753}a^{17}-\frac{1821089017299}{2232364087012}a^{16}+\frac{2693374972883}{1116182043506}a^{15}-\frac{5494934333463}{2232364087012}a^{14}+\frac{9560647682019}{2232364087012}a^{13}-\frac{4950088458799}{2232364087012}a^{12}+\frac{1520134389559}{1116182043506}a^{11}+\frac{9692358878383}{2232364087012}a^{10}-\frac{9123273747903}{1116182043506}a^{9}+\frac{20793298064745}{1116182043506}a^{8}-\frac{223079169288}{12978860971}a^{7}+\frac{75868593750281}{2232364087012}a^{6}-\frac{33896708987759}{2232364087012}a^{5}+\frac{38997700181853}{1116182043506}a^{4}-\frac{14281160073075}{2232364087012}a^{3}+\frac{10809476382723}{558091021753}a^{2}-\frac{9944030916207}{2232364087012}a+\frac{1829817931195}{1116182043506}$, $\frac{83768779750}{558091021753}a^{19}-\frac{584329444795}{2232364087012}a^{18}+\frac{1684908883153}{1116182043506}a^{17}-\frac{2516970512775}{1116182043506}a^{16}+\frac{6979553179747}{1116182043506}a^{15}-\frac{3837108783681}{558091021753}a^{14}+\frac{6416118905848}{558091021753}a^{13}-\frac{17583109866113}{2232364087012}a^{12}+\frac{16080906643467}{2232364087012}a^{11}+\frac{8859179314627}{2232364087012}a^{10}-\frac{25421910485713}{2232364087012}a^{9}+\frac{79226277843987}{2232364087012}a^{8}-\frac{1629321276257}{51915443884}a^{7}+\frac{154723875496365}{2232364087012}a^{6}-\frac{62464497856203}{2232364087012}a^{5}+\frac{164359334405569}{2232364087012}a^{4}-\frac{16222781586495}{1116182043506}a^{3}+\frac{89521710979683}{2232364087012}a^{2}-\frac{14272267689527}{2232364087012}a+\frac{9198506396227}{2232364087012}$
| 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: | \( 6484.27631918 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 6484.27631918 \cdot 2}{2\cdot\sqrt{6802209109663753569286129}}\cr\approx \mathstrut & 0.238415895898 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 14*x^18 - 33*x^17 + 85*x^16 - 156*x^15 + 281*x^14 - 417*x^13 + 590*x^12 - 734*x^11 + 869*x^10 - 875*x^9 + 908*x^8 - 679*x^7 + 668*x^6 - 321*x^5 + 303*x^4 - 77*x^3 + 42*x^2 - 4*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 - 3*x^19 + 14*x^18 - 33*x^17 + 85*x^16 - 156*x^15 + 281*x^14 - 417*x^13 + 590*x^12 - 734*x^11 + 869*x^10 - 875*x^9 + 908*x^8 - 679*x^7 + 668*x^6 - 321*x^5 + 303*x^4 - 77*x^3 + 42*x^2 - 4*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 - 3*x^19 + 14*x^18 - 33*x^17 + 85*x^16 - 156*x^15 + 281*x^14 - 417*x^13 + 590*x^12 - 734*x^11 + 869*x^10 - 875*x^9 + 908*x^8 - 679*x^7 + 668*x^6 - 321*x^5 + 303*x^4 - 77*x^3 + 42*x^2 - 4*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 - 3*x^19 + 14*x^18 - 33*x^17 + 85*x^16 - 156*x^15 + 281*x^14 - 417*x^13 + 590*x^12 - 734*x^11 + 869*x^10 - 875*x^9 + 908*x^8 - 679*x^7 + 668*x^6 - 321*x^5 + 303*x^4 - 77*x^3 + 42*x^2 - 4*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
$C_2\wr C_5$ (as 20T46):
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 160 |
| The 16 conjugacy class representatives for $C_2\wr C_5$ |
| Character table for $C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.4.4930254263.1, 10.4.2608104505127.1, 10.2.113395848049.1 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.4.4930254263.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |