Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*x - 1)
gp: K = bnfinit(y^21 - 6*y^20 - y^19 + 71*y^18 - 93*y^17 - 305*y^16 + 665*y^15 + 567*y^14 - 2100*y^13 - 214*y^12 + 3718*y^11 - 982*y^10 - 3941*y^9 + 1934*y^8 + 2506*y^7 - 1669*y^6 - 928*y^5 + 820*y^4 + 128*y^3 - 214*y^2 + 45*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*x - 1)
\( x^{21} - 6 x^{20} - x^{19} + 71 x^{18} - 93 x^{17} - 305 x^{16} + 665 x^{15} + 567 x^{14} - 2100 x^{13} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[11, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-34428615704115413560094117894967\)
\(\medspace = -\,3^{9}\cdot 280909^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.75\) | 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: | $3^{1/2}280909^{1/2}\approx 918.0016339854739$ | ||
| Ramified primes: |
\(3\), \(280909\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-842727}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{18}+\frac{1}{6}a^{17}+\frac{1}{6}a^{16}-\frac{1}{6}a^{15}+\frac{1}{6}a^{14}-\frac{1}{6}a^{13}+\frac{1}{6}a^{12}-\frac{1}{3}a^{11}+\frac{1}{6}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{1}{6}a+\frac{1}{6}$, $\frac{1}{150}a^{19}-\frac{1}{50}a^{18}+\frac{6}{25}a^{17}+\frac{8}{75}a^{16}+\frac{11}{150}a^{15}-\frac{11}{150}a^{14}+\frac{19}{75}a^{13}-\frac{1}{2}a^{12}-\frac{9}{50}a^{11}+\frac{11}{30}a^{10}-\frac{3}{50}a^{9}-\frac{2}{75}a^{8}+\frac{4}{75}a^{7}+\frac{37}{75}a^{6}+\frac{71}{150}a^{5}+\frac{73}{150}a^{4}-\frac{43}{150}a^{3}-\frac{13}{75}a^{2}-\frac{14}{75}a+\frac{28}{75}$, $\frac{1}{150}a^{20}+\frac{1}{75}a^{18}+\frac{4}{25}a^{17}+\frac{17}{75}a^{16}-\frac{14}{75}a^{15}-\frac{2}{15}a^{14}-\frac{11}{150}a^{13}+\frac{23}{150}a^{12}-\frac{17}{50}a^{11}+\frac{28}{75}a^{10}+\frac{23}{50}a^{9}-\frac{9}{25}a^{8}+\frac{73}{150}a^{7}-\frac{7}{150}a^{6}-\frac{32}{75}a^{5}+\frac{1}{150}a^{4}-\frac{11}{30}a^{3}-\frac{28}{75}a^{2}-\frac{1}{50}a+\frac{34}{75}$
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: |
$\frac{23}{30}a^{20}-\frac{659}{150}a^{19}-\frac{53}{25}a^{18}+\frac{8161}{150}a^{17}-\frac{1364}{25}a^{16}-\frac{12823}{50}a^{15}+\frac{21583}{50}a^{14}+\frac{44809}{75}a^{13}-\frac{14337}{10}a^{12}-\frac{52186}{75}a^{11}+\frac{80009}{30}a^{10}+\frac{17758}{75}a^{9}-\frac{452249}{150}a^{8}+\frac{53473}{150}a^{7}+\frac{158777}{75}a^{6}-\frac{24083}{50}a^{5}-\frac{46339}{50}a^{4}+\frac{13829}{50}a^{3}+\frac{10723}{50}a^{2}-\frac{12293}{150}a+\frac{191}{150}$, $a^{20}-5a^{19}-6a^{18}+65a^{17}-28a^{16}-333a^{15}+332a^{14}+899a^{13}-1201a^{12}-1415a^{11}+2303a^{10}+1321a^{9}-2620a^{8}-686a^{7}+1820a^{6}+151a^{5}-777a^{4}+43a^{3}+171a^{2}-43a+2$, $\frac{209}{150}a^{20}-\frac{36}{5}a^{19}-\frac{521}{75}a^{18}+\frac{13711}{150}a^{17}-\frac{2883}{50}a^{16}-\frac{11247}{25}a^{15}+\frac{1123}{2}a^{14}+\frac{85043}{75}a^{13}-\frac{294043}{150}a^{12}-\frac{117787}{75}a^{11}+\frac{279277}{75}a^{10}+\frac{164441}{150}a^{9}-\frac{637891}{150}a^{8}-\frac{3418}{25}a^{7}+\frac{224771}{75}a^{6}-\frac{8326}{25}a^{5}-\frac{65377}{50}a^{4}+\frac{1394}{5}a^{3}+\frac{14917}{50}a^{2}-\frac{7781}{75}a+\frac{407}{150}$, $\frac{37}{75}a^{20}-\frac{104}{75}a^{19}-\frac{351}{50}a^{18}+\frac{3113}{150}a^{17}+\frac{1048}{25}a^{16}-\frac{1289}{10}a^{15}-\frac{3607}{25}a^{14}+\frac{33266}{75}a^{13}+\frac{8142}{25}a^{12}-\frac{140683}{150}a^{11}-\frac{76021}{150}a^{10}+\frac{95114}{75}a^{9}+\frac{81461}{150}a^{8}-\frac{166237}{150}a^{7}-\frac{11747}{30}a^{6}+\frac{15516}{25}a^{5}+\frac{1851}{10}a^{4}-\frac{10467}{50}a^{3}-\frac{2437}{50}a^{2}+\frac{4727}{150}a+\frac{109}{150}$, $\frac{22}{25}a^{20}-\frac{829}{150}a^{19}+\frac{1}{150}a^{18}+\frac{4937}{75}a^{17}-\frac{2396}{25}a^{16}-\frac{1433}{5}a^{15}+\frac{33893}{50}a^{14}+\frac{27657}{50}a^{13}-\frac{164257}{75}a^{12}-\frac{22012}{75}a^{11}+\frac{101312}{25}a^{10}-\frac{106381}{150}a^{9}-\frac{115252}{25}a^{8}+\frac{228029}{150}a^{7}+\frac{49198}{15}a^{6}-\frac{65819}{50}a^{5}-\frac{14599}{10}a^{4}+\frac{31929}{50}a^{3}+\frac{8652}{25}a^{2}-\frac{8053}{50}a+\frac{226}{75}$, $\frac{11}{25}a^{20}-\frac{151}{75}a^{19}-\frac{487}{150}a^{18}+\frac{1981}{75}a^{17}-\frac{98}{25}a^{16}-\frac{1373}{10}a^{15}+\frac{4959}{50}a^{14}+\frac{9483}{25}a^{13}-\frac{29066}{75}a^{12}-\frac{46556}{75}a^{11}+\frac{19031}{25}a^{10}+\frac{93697}{150}a^{9}-\frac{43727}{50}a^{8}-\frac{28999}{75}a^{7}+\frac{18283}{30}a^{6}+\frac{3639}{25}a^{5}-\frac{2597}{10}a^{4}-\frac{524}{25}a^{3}+\frac{2877}{50}a^{2}-\frac{182}{25}a+\frac{1}{150}$, $\frac{47}{150}a^{20}-\frac{91}{50}a^{19}-\frac{137}{150}a^{18}+23a^{17}-\frac{659}{30}a^{16}-\frac{16919}{150}a^{15}+\frac{26513}{150}a^{14}+\frac{21442}{75}a^{13}-\frac{44572}{75}a^{12}-\frac{10321}{25}a^{11}+\frac{83921}{75}a^{10}+\frac{691}{2}a^{9}-\frac{63957}{50}a^{8}-\frac{24253}{150}a^{7}+\frac{67772}{75}a^{6}+\frac{6259}{150}a^{5}-\frac{29366}{75}a^{4}+\frac{227}{75}a^{3}+\frac{12941}{150}a^{2}-\frac{449}{50}a-\frac{17}{150}$, $\frac{1}{150}a^{20}+\frac{2}{5}a^{19}-\frac{101}{50}a^{18}-\frac{208}{75}a^{17}+\frac{3869}{150}a^{16}-\frac{893}{150}a^{15}-\frac{1993}{15}a^{14}+\frac{2574}{25}a^{13}+\frac{9133}{25}a^{12}-\frac{29173}{75}a^{11}-\frac{14899}{25}a^{10}+\frac{56602}{75}a^{9}+\frac{44878}{75}a^{8}-\frac{129247}{150}a^{7}-\frac{27746}{75}a^{6}+\frac{44873}{75}a^{5}+\frac{10453}{75}a^{4}-\frac{7627}{30}a^{3}-\frac{1508}{75}a^{2}+\frac{8417}{150}a-\frac{349}{50}$, $\frac{3}{10}a^{20}-\frac{319}{150}a^{19}+\frac{12}{25}a^{18}+\frac{1307}{50}a^{17}-\frac{2987}{75}a^{16}-\frac{9097}{75}a^{15}+\frac{20617}{75}a^{14}+\frac{40933}{150}a^{13}-\frac{8921}{10}a^{12}-\frac{7222}{25}a^{11}+\frac{10031}{6}a^{10}+\frac{1767}{50}a^{9}-\frac{290129}{150}a^{8}+\frac{18629}{75}a^{7}+\frac{210679}{150}a^{6}-\frac{40979}{150}a^{5}-\frac{95917}{150}a^{4}+\frac{21367}{150}a^{3}+\frac{23849}{150}a^{2}-\frac{2914}{75}a-\frac{277}{75}$, $\frac{149}{75}a^{20}-\frac{1349}{150}a^{19}-\frac{1166}{75}a^{18}+\frac{5971}{50}a^{17}-\frac{877}{150}a^{16}-\frac{47404}{75}a^{15}+\frac{57629}{150}a^{14}+\frac{17889}{10}a^{13}-\frac{234571}{150}a^{12}-3012a^{11}+\frac{155131}{50}a^{10}+\frac{77838}{25}a^{9}-\frac{267398}{75}a^{8}-\frac{97371}{50}a^{7}+\frac{185819}{75}a^{6}+\frac{105949}{150}a^{5}-\frac{78902}{75}a^{4}-\frac{6454}{75}a^{3}+\frac{17368}{75}a^{2}-\frac{6697}{150}a+\frac{22}{15}$, $\frac{32}{75}a^{20}-\frac{38}{25}a^{19}-\frac{763}{150}a^{18}+\frac{1101}{50}a^{17}+\frac{1664}{75}a^{16}-\frac{787}{6}a^{15}-\frac{3286}{75}a^{14}+\frac{32291}{75}a^{13}+\frac{2461}{75}a^{12}-\frac{43411}{50}a^{11}+\frac{2069}{150}a^{10}+\frac{28053}{25}a^{9}-\frac{1723}{50}a^{8}-\frac{140927}{150}a^{7}+\frac{271}{30}a^{6}+\frac{38158}{75}a^{5}+\frac{419}{30}a^{4}-\frac{24841}{150}a^{3}-\frac{1931}{150}a^{2}+\frac{889}{50}a+\frac{209}{150}$, $\frac{13}{30}a^{20}-\frac{329}{150}a^{19}-\frac{179}{75}a^{18}+\frac{2108}{75}a^{17}-\frac{1102}{75}a^{16}-\frac{21139}{150}a^{15}+\frac{11672}{75}a^{14}+\frac{9218}{25}a^{13}-\frac{16451}{30}a^{12}-\frac{82907}{150}a^{11}+\frac{5184}{5}a^{10}+\frac{70721}{150}a^{9}-\frac{175469}{150}a^{8}-\frac{14531}{75}a^{7}+\frac{20279}{25}a^{6}-\frac{19}{150}a^{5}-\frac{52277}{150}a^{4}+\frac{3286}{75}a^{3}+\frac{11689}{150}a^{2}-\frac{1729}{75}a-\frac{29}{150}$, $\frac{427}{150}a^{20}-\frac{2213}{150}a^{19}-\frac{347}{25}a^{18}+\frac{2803}{15}a^{17}-\frac{613}{5}a^{16}-\frac{45733}{50}a^{15}+\frac{29413}{25}a^{14}+\frac{170767}{75}a^{13}-\frac{204993}{50}a^{12}-\frac{459551}{150}a^{11}+\frac{584411}{75}a^{10}+\frac{58591}{30}a^{9}-\frac{1337531}{150}a^{8}+\frac{4196}{75}a^{7}+\frac{472937}{75}a^{6}-\frac{45767}{50}a^{5}-\frac{138549}{50}a^{4}+\frac{16554}{25}a^{3}+\frac{32267}{50}a^{2}-\frac{16771}{75}a+\frac{383}{150}$, $\frac{19}{15}a^{20}-\frac{279}{50}a^{19}-\frac{503}{50}a^{18}+\frac{11053}{150}a^{17}-\frac{191}{75}a^{16}-\frac{28876}{75}a^{15}+\frac{18316}{75}a^{14}+\frac{53093}{50}a^{13}-\frac{5076}{5}a^{12}-\frac{256091}{150}a^{11}+\frac{20567}{10}a^{10}+\frac{122284}{75}a^{9}-\frac{182006}{75}a^{8}-\frac{66038}{75}a^{7}+\frac{130591}{75}a^{6}+\frac{33463}{150}a^{5}-\frac{115561}{150}a^{4}+\frac{2483}{75}a^{3}+\frac{13861}{75}a^{2}-\frac{3242}{75}a-\frac{42}{25}$, $\frac{377}{150}a^{20}-\frac{1091}{75}a^{19}-6a^{18}+\frac{13423}{75}a^{17}-\frac{9623}{50}a^{16}-\frac{20793}{25}a^{15}+\frac{74679}{50}a^{14}+\frac{140606}{75}a^{13}-\frac{123584}{25}a^{12}-\frac{148694}{75}a^{11}+\frac{689626}{75}a^{10}+\frac{30751}{150}a^{9}-\frac{312071}{30}a^{8}+\frac{54373}{30}a^{7}+\frac{1094243}{150}a^{6}-2018a^{5}-\frac{158703}{50}a^{4}+\frac{54397}{50}a^{3}+\frac{7223}{10}a^{2}-\frac{9137}{30}a+\frac{947}{75}$
| 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: | \( 130916585.101 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{5}\cdot 130916585.101 \cdot 1}{2\cdot\sqrt{34428615704115413560094117894967}}\cr\approx \mathstrut & 0.223735132919 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*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^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*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^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*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^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*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
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 5040 |
| The 15 conjugacy class representatives for $S_7$ |
| Character table for $S_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 7 sibling: | 7.5.7584543.1 |
| Degree 14 sibling: | deg 14 |
| Degree 30 sibling: | data not computed |
| Degree 35 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | 7.5.7584543.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{3}$ | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.7.0.1}{7} }^{3}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.7.0.1}{7} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(280909\)
| $\Q_{280909}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $2$ | $4$ | $4$ |