Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 7*x^20 - 3*x^19 + 17*x^18 + 12*x^17 - 15*x^16 - 16*x^15 - x^14 - x^13 - 28*x^12 - 42*x^11 + 3*x^10 + 25*x^9 - 16*x^8 - 47*x^7 - 35*x^6 + 7*x^5 + 16*x^4 - 7*x^3 - 5*x^2 + 2*x + 1)
gp: K = bnfinit(y^22 - 7*y^20 - 3*y^19 + 17*y^18 + 12*y^17 - 15*y^16 - 16*y^15 - y^14 - y^13 - 28*y^12 - 42*y^11 + 3*y^10 + 25*y^9 - 16*y^8 - 47*y^7 - 35*y^6 + 7*y^5 + 16*y^4 - 7*y^3 - 5*y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 7*x^20 - 3*x^19 + 17*x^18 + 12*x^17 - 15*x^16 - 16*x^15 - x^14 - x^13 - 28*x^12 - 42*x^11 + 3*x^10 + 25*x^9 - 16*x^8 - 47*x^7 - 35*x^6 + 7*x^5 + 16*x^4 - 7*x^3 - 5*x^2 + 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 7*x^20 - 3*x^19 + 17*x^18 + 12*x^17 - 15*x^16 - 16*x^15 - x^14 - x^13 - 28*x^12 - 42*x^11 + 3*x^10 + 25*x^9 - 16*x^8 - 47*x^7 - 35*x^6 + 7*x^5 + 16*x^4 - 7*x^3 - 5*x^2 + 2*x + 1)
\( x^{22} - 7 x^{20} - 3 x^{19} + 17 x^{18} + 12 x^{17} - 15 x^{16} - 16 x^{15} - x^{14} - x^{13} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(41217344982318472393444237\)
\(\medspace = 1070197\cdot 6205948139^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.60\) | 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: | $1070197^{1/2}6205948139^{1/2}\approx 81495932.90780456$ | ||
| Ramified primes: |
\(1070197\), \(6205948139\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1070197}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{286622459888549}a^{21}+\frac{124116692696897}{286622459888549}a^{20}+\frac{136340077927243}{286622459888549}a^{19}+\frac{121235386306813}{286622459888549}a^{18}+\frac{89595491501273}{286622459888549}a^{17}-\frac{33027992564560}{286622459888549}a^{16}-\frac{87369984719483}{286622459888549}a^{15}-\frac{122809513073835}{286622459888549}a^{14}+\frac{90884654961141}{286622459888549}a^{13}+\frac{29847846066195}{286622459888549}a^{12}+\frac{40414001380659}{286622459888549}a^{11}-\frac{44873071692605}{286622459888549}a^{10}+\frac{55513582153561}{286622459888549}a^{9}-\frac{46279267809093}{286622459888549}a^{8}+\frac{38113788694250}{286622459888549}a^{7}+\frac{38417120769084}{286622459888549}a^{6}+\frac{81447494381516}{286622459888549}a^{5}-\frac{87334516585045}{286622459888549}a^{4}-\frac{58602607821414}{286622459888549}a^{3}+\frac{62771417801988}{286622459888549}a^{2}-\frac{33988661821785}{286622459888549}a-\frac{51407076939844}{286622459888549}$
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: | $11$ | 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: |
$\frac{168451973026231}{286622459888549}a^{21}-\frac{21196155271796}{286622459888549}a^{20}-\frac{11\!\cdots\!33}{286622459888549}a^{19}-\frac{318918840878670}{286622459888549}a^{18}+\frac{30\!\cdots\!81}{286622459888549}a^{17}+\frac{14\!\cdots\!93}{286622459888549}a^{16}-\frac{29\!\cdots\!81}{286622459888549}a^{15}-\frac{20\!\cdots\!77}{286622459888549}a^{14}+\frac{375631659639492}{286622459888549}a^{13}-\frac{375618860395486}{286622459888549}a^{12}-\frac{47\!\cdots\!44}{286622459888549}a^{11}-\frac{63\!\cdots\!04}{286622459888549}a^{10}+\frac{19\!\cdots\!13}{286622459888549}a^{9}+\frac{40\!\cdots\!25}{286622459888549}a^{8}-\frac{37\!\cdots\!91}{286622459888549}a^{7}-\frac{70\!\cdots\!72}{286622459888549}a^{6}-\frac{41\!\cdots\!04}{286622459888549}a^{5}+\frac{21\!\cdots\!46}{286622459888549}a^{4}+\frac{22\!\cdots\!66}{286622459888549}a^{3}-\frac{17\!\cdots\!37}{286622459888549}a^{2}-\frac{142732789873595}{286622459888549}a+\frac{610009943635148}{286622459888549}$, $\frac{229282145389378}{286622459888549}a^{21}-\frac{130463614705039}{286622459888549}a^{20}-\frac{15\!\cdots\!96}{286622459888549}a^{19}+\frac{124219490715321}{286622459888549}a^{18}+\frac{36\!\cdots\!58}{286622459888549}a^{17}+\frac{873303411213974}{286622459888549}a^{16}-\frac{34\!\cdots\!48}{286622459888549}a^{15}-\frac{20\!\cdots\!69}{286622459888549}a^{14}+\frac{244731808393220}{286622459888549}a^{13}-\frac{376394715003531}{286622459888549}a^{12}-\frac{57\!\cdots\!24}{286622459888549}a^{11}-\frac{61\!\cdots\!87}{286622459888549}a^{10}+\frac{36\!\cdots\!00}{286622459888549}a^{9}+\frac{39\!\cdots\!67}{286622459888549}a^{8}-\frac{43\!\cdots\!00}{286622459888549}a^{7}-\frac{78\!\cdots\!67}{286622459888549}a^{6}-\frac{41\!\cdots\!37}{286622459888549}a^{5}+\frac{34\!\cdots\!82}{286622459888549}a^{4}+\frac{25\!\cdots\!93}{286622459888549}a^{3}-\frac{13\!\cdots\!14}{286622459888549}a^{2}+\frac{80855785138363}{286622459888549}a+\frac{242364853749620}{286622459888549}$, $\frac{135433006666015}{286622459888549}a^{21}+\frac{3312216398386}{286622459888549}a^{20}-\frac{941282410634547}{286622459888549}a^{19}-\frac{365116632775892}{286622459888549}a^{18}+\frac{21\!\cdots\!45}{286622459888549}a^{17}+\frac{13\!\cdots\!28}{286622459888549}a^{16}-\frac{15\!\cdots\!13}{286622459888549}a^{15}-\frac{14\!\cdots\!21}{286622459888549}a^{14}-\frac{595983872666292}{286622459888549}a^{13}-\frac{802465461030408}{286622459888549}a^{12}-\frac{39\!\cdots\!20}{286622459888549}a^{11}-\frac{57\!\cdots\!25}{286622459888549}a^{10}-\frac{83637015274109}{286622459888549}a^{9}+\frac{16\!\cdots\!08}{286622459888549}a^{8}-\frac{25\!\cdots\!13}{286622459888549}a^{7}-\frac{51\!\cdots\!31}{286622459888549}a^{6}-\frac{51\!\cdots\!40}{286622459888549}a^{5}-\frac{392489861654566}{286622459888549}a^{4}+\frac{597101860352844}{286622459888549}a^{3}-\frac{12\!\cdots\!13}{286622459888549}a^{2}+\frac{105828673948907}{286622459888549}a-\frac{31983592776548}{286622459888549}$, $\frac{202243604667807}{286622459888549}a^{21}-\frac{123452380647776}{286622459888549}a^{20}-\frac{13\!\cdots\!98}{286622459888549}a^{19}+\frac{287721349792363}{286622459888549}a^{18}+\frac{34\!\cdots\!64}{286622459888549}a^{17}+\frac{89406145218174}{286622459888549}a^{16}-\frac{36\!\cdots\!15}{286622459888549}a^{15}-\frac{718883323615151}{286622459888549}a^{14}+\frac{852525686303227}{286622459888549}a^{13}-\frac{812591099485696}{286622459888549}a^{12}-\frac{55\!\cdots\!53}{286622459888549}a^{11}-\frac{50\!\cdots\!37}{286622459888549}a^{10}+\frac{47\!\cdots\!09}{286622459888549}a^{9}+\frac{24\!\cdots\!33}{286622459888549}a^{8}-\frac{58\!\cdots\!41}{286622459888549}a^{7}-\frac{58\!\cdots\!71}{286622459888549}a^{6}-\frac{23\!\cdots\!55}{286622459888549}a^{5}+\frac{38\!\cdots\!46}{286622459888549}a^{4}+\frac{791589882266682}{286622459888549}a^{3}-\frac{28\!\cdots\!38}{286622459888549}a^{2}+\frac{899393432121374}{286622459888549}a+\frac{378351480089373}{286622459888549}$, $\frac{291408130209244}{286622459888549}a^{21}-\frac{65244891438514}{286622459888549}a^{20}-\frac{20\!\cdots\!52}{286622459888549}a^{19}-\frac{361927513102903}{286622459888549}a^{18}+\frac{49\!\cdots\!14}{286622459888549}a^{17}+\frac{20\!\cdots\!15}{286622459888549}a^{16}-\frac{46\!\cdots\!90}{286622459888549}a^{15}-\frac{29\!\cdots\!68}{286622459888549}a^{14}+\frac{258374359449610}{286622459888549}a^{13}-\frac{879202597551884}{286622459888549}a^{12}-\frac{81\!\cdots\!98}{286622459888549}a^{11}-\frac{10\!\cdots\!11}{286622459888549}a^{10}+\frac{31\!\cdots\!34}{286622459888549}a^{9}+\frac{54\!\cdots\!83}{286622459888549}a^{8}-\frac{66\!\cdots\!09}{286622459888549}a^{7}-\frac{11\!\cdots\!34}{286622459888549}a^{6}-\frac{73\!\cdots\!39}{286622459888549}a^{5}+\frac{29\!\cdots\!04}{286622459888549}a^{4}+\frac{26\!\cdots\!48}{286622459888549}a^{3}-\frac{31\!\cdots\!00}{286622459888549}a^{2}-\frac{187097050611542}{286622459888549}a+\frac{636300466274048}{286622459888549}$, $a$, $\frac{238465518548486}{286622459888549}a^{21}-\frac{154824899137666}{286622459888549}a^{20}-\frac{16\!\cdots\!22}{286622459888549}a^{19}+\frac{412125904459590}{286622459888549}a^{18}+\frac{43\!\cdots\!80}{286622459888549}a^{17}+\frac{10166259118809}{286622459888549}a^{16}-\frac{50\!\cdots\!93}{286622459888549}a^{15}-\frac{10\!\cdots\!61}{286622459888549}a^{14}+\frac{17\!\cdots\!10}{286622459888549}a^{13}-\frac{322734136908782}{286622459888549}a^{12}-\frac{62\!\cdots\!29}{286622459888549}a^{11}-\frac{58\!\cdots\!31}{286622459888549}a^{10}+\frac{65\!\cdots\!94}{286622459888549}a^{9}+\frac{43\!\cdots\!43}{286622459888549}a^{8}-\frac{77\!\cdots\!80}{286622459888549}a^{7}-\frac{79\!\cdots\!24}{286622459888549}a^{6}-\frac{16\!\cdots\!63}{286622459888549}a^{5}+\frac{61\!\cdots\!10}{286622459888549}a^{4}+\frac{21\!\cdots\!53}{286622459888549}a^{3}-\frac{40\!\cdots\!68}{286622459888549}a^{2}+\frac{29775752537938}{286622459888549}a+\frac{762868388946196}{286622459888549}$, $\frac{291408130209244}{286622459888549}a^{21}-\frac{65244891438514}{286622459888549}a^{20}-\frac{20\!\cdots\!52}{286622459888549}a^{19}-\frac{361927513102903}{286622459888549}a^{18}+\frac{49\!\cdots\!14}{286622459888549}a^{17}+\frac{20\!\cdots\!15}{286622459888549}a^{16}-\frac{46\!\cdots\!90}{286622459888549}a^{15}-\frac{29\!\cdots\!68}{286622459888549}a^{14}+\frac{258374359449610}{286622459888549}a^{13}-\frac{879202597551884}{286622459888549}a^{12}-\frac{81\!\cdots\!98}{286622459888549}a^{11}-\frac{10\!\cdots\!11}{286622459888549}a^{10}+\frac{31\!\cdots\!34}{286622459888549}a^{9}+\frac{54\!\cdots\!83}{286622459888549}a^{8}-\frac{66\!\cdots\!09}{286622459888549}a^{7}-\frac{11\!\cdots\!34}{286622459888549}a^{6}-\frac{73\!\cdots\!39}{286622459888549}a^{5}+\frac{29\!\cdots\!04}{286622459888549}a^{4}+\frac{26\!\cdots\!48}{286622459888549}a^{3}-\frac{31\!\cdots\!00}{286622459888549}a^{2}+\frac{99525409277007}{286622459888549}a+\frac{636300466274048}{286622459888549}$, $\frac{581768070926643}{286622459888549}a^{21}-\frac{268487418329054}{286622459888549}a^{20}-\frac{38\!\cdots\!52}{286622459888549}a^{19}-\frac{35696618374389}{286622459888549}a^{18}+\frac{93\!\cdots\!04}{286622459888549}a^{17}+\frac{27\!\cdots\!72}{286622459888549}a^{16}-\frac{87\!\cdots\!79}{286622459888549}a^{15}-\frac{49\!\cdots\!13}{286622459888549}a^{14}+\frac{710198743232533}{286622459888549}a^{13}-\frac{16\!\cdots\!57}{286622459888549}a^{12}-\frac{15\!\cdots\!62}{286622459888549}a^{11}-\frac{17\!\cdots\!61}{286622459888549}a^{10}+\frac{73\!\cdots\!12}{286622459888549}a^{9}+\frac{89\!\cdots\!86}{286622459888549}a^{8}-\frac{12\!\cdots\!55}{286622459888549}a^{7}-\frac{20\!\cdots\!94}{286622459888549}a^{6}-\frac{12\!\cdots\!92}{286622459888549}a^{5}+\frac{64\!\cdots\!51}{286622459888549}a^{4}+\frac{44\!\cdots\!86}{286622459888549}a^{3}-\frac{54\!\cdots\!29}{286622459888549}a^{2}-\frac{263579214457657}{286622459888549}a+\frac{11\!\cdots\!32}{286622459888549}$, $\frac{11825138737}{286622459888549}a^{21}-\frac{15368320375513}{286622459888549}a^{20}+\frac{26717344645613}{286622459888549}a^{19}+\frac{125228055275214}{286622459888549}a^{18}-\frac{128022891504438}{286622459888549}a^{17}-\frac{458267775431677}{286622459888549}a^{16}+\frac{113855306748009}{286622459888549}a^{15}+\frac{768485639372013}{286622459888549}a^{14}+\frac{317405044143745}{286622459888549}a^{13}-\frac{369148257833540}{286622459888549}a^{12}-\frac{530799879110077}{286622459888549}a^{11}-\frac{22129805074501}{286622459888549}a^{10}-\frac{185123203678223}{286622459888549}a^{9}-\frac{17\!\cdots\!18}{286622459888549}a^{8}-\frac{13\!\cdots\!28}{286622459888549}a^{7}+\frac{554018447174148}{286622459888549}a^{6}+\frac{795489557270836}{286622459888549}a^{5}-\frac{354339988921630}{286622459888549}a^{4}-\frac{17\!\cdots\!88}{286622459888549}a^{3}-\frac{13\!\cdots\!61}{286622459888549}a^{2}+\frac{181676150279612}{286622459888549}a+\frac{158165914603406}{286622459888549}$, $\frac{57479633730908}{286622459888549}a^{21}-\frac{44031512180530}{286622459888549}a^{20}-\frac{272462632263173}{286622459888549}a^{19}-\frac{664899090507}{286622459888549}a^{18}+\frac{360083924660248}{286622459888549}a^{17}+\frac{314185822326099}{286622459888549}a^{16}+\frac{336029394905718}{286622459888549}a^{15}-\frac{426060395158133}{286622459888549}a^{14}-\frac{948053150824613}{286622459888549}a^{13}-\frac{487690429061548}{286622459888549}a^{12}-\frac{12\!\cdots\!42}{286622459888549}a^{11}-\frac{17\!\cdots\!57}{286622459888549}a^{10}-\frac{12\!\cdots\!16}{286622459888549}a^{9}-\frac{526591647211655}{286622459888549}a^{8}+\frac{167580741901273}{286622459888549}a^{7}-\frac{14\!\cdots\!72}{286622459888549}a^{6}-\frac{22\!\cdots\!93}{286622459888549}a^{5}-\frac{12\!\cdots\!48}{286622459888549}a^{4}-\frac{530801690240694}{286622459888549}a^{3}+\frac{197650926725560}{286622459888549}a^{2}+\frac{140986844696695}{286622459888549}a-\frac{123480724665294}{286622459888549}$
| 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: | \( 5727.32762159 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 5727.32762159 \cdot 1}{2\cdot\sqrt{41217344982318472393444237}}\cr\approx \mathstrut & 0.171096332138 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 7*x^20 - 3*x^19 + 17*x^18 + 12*x^17 - 15*x^16 - 16*x^15 - x^14 - x^13 - 28*x^12 - 42*x^11 + 3*x^10 + 25*x^9 - 16*x^8 - 47*x^7 - 35*x^6 + 7*x^5 + 16*x^4 - 7*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^22 - 7*x^20 - 3*x^19 + 17*x^18 + 12*x^17 - 15*x^16 - 16*x^15 - x^14 - x^13 - 28*x^12 - 42*x^11 + 3*x^10 + 25*x^9 - 16*x^8 - 47*x^7 - 35*x^6 + 7*x^5 + 16*x^4 - 7*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^22 - 7*x^20 - 3*x^19 + 17*x^18 + 12*x^17 - 15*x^16 - 16*x^15 - x^14 - x^13 - 28*x^12 - 42*x^11 + 3*x^10 + 25*x^9 - 16*x^8 - 47*x^7 - 35*x^6 + 7*x^5 + 16*x^4 - 7*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^22 - 7*x^20 - 3*x^19 + 17*x^18 + 12*x^17 - 15*x^16 - 16*x^15 - x^14 - x^13 - 28*x^12 - 42*x^11 + 3*x^10 + 25*x^9 - 16*x^8 - 47*x^7 - 35*x^6 + 7*x^5 + 16*x^4 - 7*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
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
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 81749606400 |
| The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ |
| Character table for $C_2^{10}.(C_2\times S_{11})$ |
Intermediate fields
| 11.1.6205948139.1 |
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 22 sibling: | data not computed |
| Degree 44 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/23.10.0.1}{10} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.7.0.1}{7} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1070197\)
| $\Q_{1070197}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1070197}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
|
\(6205948139\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
| Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |