Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 100*x^10 + 96*x^9 + 3746*x^8 - 3362*x^7 - 65339*x^6 + 51584*x^5 + 525352*x^4 - 322117*x^3 - 1551355*x^2 + 579413*x + 104959)
gp: K = bnfinit(y^12 - y^11 - 100*y^10 + 96*y^9 + 3746*y^8 - 3362*y^7 - 65339*y^6 + 51584*y^5 + 525352*y^4 - 322117*y^3 - 1551355*y^2 + 579413*y + 104959, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 100*x^10 + 96*x^9 + 3746*x^8 - 3362*x^7 - 65339*x^6 + 51584*x^5 + 525352*x^4 - 322117*x^3 - 1551355*x^2 + 579413*x + 104959);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 100*x^10 + 96*x^9 + 3746*x^8 - 3362*x^7 - 65339*x^6 + 51584*x^5 + 525352*x^4 - 322117*x^3 - 1551355*x^2 + 579413*x + 104959)
\( x^{12} - x^{11} - 100 x^{10} + 96 x^{9} + 3746 x^{8} - 3362 x^{7} - 65339 x^{6} + 51584 x^{5} + \cdots + 104959 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(24420144017624194286937\)
\(\medspace = 3^{6}\cdot 7^{10}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(73.39\) | 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: | $3^{1/2}7^{5/6}17^{3/4}\approx 73.39148665417294$ | ||
| Ramified primes: |
\(3\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(357=3\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{357}(256,·)$, $\chi_{357}(1,·)$, $\chi_{357}(67,·)$, $\chi_{357}(293,·)$, $\chi_{357}(38,·)$, $\chi_{357}(353,·)$, $\chi_{357}(169,·)$, $\chi_{357}(205,·)$, $\chi_{357}(47,·)$, $\chi_{357}(16,·)$, $\chi_{357}(89,·)$, $\chi_{357}(251,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{473476}a^{10}+\frac{790}{118369}a^{9}+\frac{53811}{236738}a^{8}-\frac{112683}{473476}a^{7}-\frac{51897}{473476}a^{6}-\frac{200039}{473476}a^{5}-\frac{1861}{473476}a^{4}+\frac{66967}{236738}a^{3}+\frac{235643}{473476}a^{2}+\frac{28423}{473476}a-\frac{36617}{118369}$, $\frac{1}{27\!\cdots\!52}a^{11}-\frac{6014691891663}{27\!\cdots\!52}a^{10}+\frac{15\!\cdots\!77}{27\!\cdots\!52}a^{9}+\frac{72\!\cdots\!17}{27\!\cdots\!52}a^{8}-\frac{21\!\cdots\!21}{13\!\cdots\!26}a^{7}+\frac{22\!\cdots\!41}{27\!\cdots\!52}a^{6}-\frac{64\!\cdots\!81}{27\!\cdots\!52}a^{5}+\frac{46\!\cdots\!95}{13\!\cdots\!26}a^{4}-\frac{14\!\cdots\!14}{68\!\cdots\!13}a^{3}+\frac{26\!\cdots\!17}{13\!\cdots\!26}a^{2}-\frac{26\!\cdots\!12}{68\!\cdots\!13}a+\frac{76\!\cdots\!51}{27\!\cdots\!52}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (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: |
$\frac{62587958002}{27\!\cdots\!63}a^{11}-\frac{681910289063}{10\!\cdots\!52}a^{10}-\frac{5721652586889}{27\!\cdots\!63}a^{9}+\frac{13721522902611}{27\!\cdots\!63}a^{8}+\frac{716886842913163}{10\!\cdots\!52}a^{7}-\frac{15\!\cdots\!97}{10\!\cdots\!52}a^{6}-\frac{86\!\cdots\!31}{10\!\cdots\!52}a^{5}+\frac{18\!\cdots\!67}{10\!\cdots\!52}a^{4}+\frac{14\!\cdots\!13}{54\!\cdots\!26}a^{3}-\frac{90\!\cdots\!45}{10\!\cdots\!52}a^{2}+\frac{26\!\cdots\!77}{10\!\cdots\!52}a+\frac{12\!\cdots\!65}{54\!\cdots\!26}$, $\frac{638565204327}{10\!\cdots\!52}a^{11}+\frac{1842419548911}{54\!\cdots\!26}a^{10}-\frac{13645457463972}{27\!\cdots\!63}a^{9}-\frac{310918947916583}{10\!\cdots\!52}a^{8}+\frac{16\!\cdots\!71}{10\!\cdots\!52}a^{7}+\frac{91\!\cdots\!53}{10\!\cdots\!52}a^{6}-\frac{21\!\cdots\!13}{10\!\cdots\!52}a^{5}-\frac{27\!\cdots\!24}{27\!\cdots\!63}a^{4}+\frac{12\!\cdots\!49}{10\!\cdots\!52}a^{3}+\frac{48\!\cdots\!53}{10\!\cdots\!52}a^{2}-\frac{47\!\cdots\!76}{27\!\cdots\!63}a-\frac{57\!\cdots\!94}{27\!\cdots\!63}$, $\frac{96651217713333}{68\!\cdots\!13}a^{11}+\frac{973533323422581}{13\!\cdots\!26}a^{10}-\frac{35\!\cdots\!27}{27\!\cdots\!52}a^{9}-\frac{82\!\cdots\!73}{13\!\cdots\!26}a^{8}+\frac{58\!\cdots\!99}{13\!\cdots\!26}a^{7}+\frac{48\!\cdots\!53}{27\!\cdots\!52}a^{6}-\frac{18\!\cdots\!81}{27\!\cdots\!52}a^{5}-\frac{58\!\cdots\!75}{27\!\cdots\!52}a^{4}+\frac{14\!\cdots\!37}{27\!\cdots\!52}a^{3}+\frac{11\!\cdots\!59}{13\!\cdots\!26}a^{2}-\frac{43\!\cdots\!79}{27\!\cdots\!52}a-\frac{68\!\cdots\!33}{27\!\cdots\!52}$, $\frac{12593184286065}{13\!\cdots\!26}a^{11}-\frac{18\!\cdots\!53}{27\!\cdots\!52}a^{10}-\frac{47\!\cdots\!69}{27\!\cdots\!52}a^{9}+\frac{38\!\cdots\!63}{68\!\cdots\!13}a^{8}+\frac{25\!\cdots\!51}{27\!\cdots\!52}a^{7}-\frac{10\!\cdots\!64}{68\!\cdots\!13}a^{6}-\frac{13\!\cdots\!45}{68\!\cdots\!13}a^{5}+\frac{24\!\cdots\!11}{13\!\cdots\!26}a^{4}+\frac{44\!\cdots\!67}{27\!\cdots\!52}a^{3}-\frac{20\!\cdots\!89}{27\!\cdots\!52}a^{2}-\frac{29\!\cdots\!30}{68\!\cdots\!13}a+\frac{11\!\cdots\!11}{27\!\cdots\!52}$, $\frac{506025967430361}{27\!\cdots\!52}a^{11}+\frac{900303082569213}{27\!\cdots\!52}a^{10}-\frac{45\!\cdots\!33}{27\!\cdots\!52}a^{9}-\frac{82\!\cdots\!47}{27\!\cdots\!52}a^{8}+\frac{75\!\cdots\!57}{13\!\cdots\!26}a^{7}+\frac{26\!\cdots\!53}{27\!\cdots\!52}a^{6}-\frac{21\!\cdots\!69}{27\!\cdots\!52}a^{5}-\frac{17\!\cdots\!33}{13\!\cdots\!26}a^{4}+\frac{34\!\cdots\!91}{68\!\cdots\!13}a^{3}+\frac{39\!\cdots\!05}{68\!\cdots\!13}a^{2}-\frac{64\!\cdots\!58}{68\!\cdots\!13}a-\frac{31\!\cdots\!65}{27\!\cdots\!52}$, $\frac{288479005446863}{27\!\cdots\!52}a^{11}-\frac{29\!\cdots\!17}{27\!\cdots\!52}a^{10}-\frac{11\!\cdots\!69}{13\!\cdots\!26}a^{9}+\frac{24\!\cdots\!99}{27\!\cdots\!52}a^{8}+\frac{34\!\cdots\!35}{13\!\cdots\!26}a^{7}-\frac{35\!\cdots\!77}{13\!\cdots\!26}a^{6}-\frac{52\!\cdots\!25}{13\!\cdots\!26}a^{5}+\frac{86\!\cdots\!25}{27\!\cdots\!52}a^{4}+\frac{98\!\cdots\!57}{27\!\cdots\!52}a^{3}-\frac{10\!\cdots\!57}{68\!\cdots\!13}a^{2}-\frac{46\!\cdots\!99}{27\!\cdots\!52}a+\frac{80\!\cdots\!48}{68\!\cdots\!13}$, $\frac{13997623211407}{27\!\cdots\!52}a^{11}-\frac{563535571281080}{68\!\cdots\!13}a^{10}+\frac{20\!\cdots\!41}{13\!\cdots\!26}a^{9}+\frac{17\!\cdots\!23}{27\!\cdots\!52}a^{8}-\frac{36\!\cdots\!65}{27\!\cdots\!52}a^{7}-\frac{46\!\cdots\!73}{27\!\cdots\!52}a^{6}+\frac{91\!\cdots\!13}{27\!\cdots\!52}a^{5}+\frac{23\!\cdots\!41}{13\!\cdots\!26}a^{4}-\frac{77\!\cdots\!63}{27\!\cdots\!52}a^{3}-\frac{14\!\cdots\!83}{27\!\cdots\!52}a^{2}+\frac{25\!\cdots\!04}{68\!\cdots\!13}a-\frac{35\!\cdots\!83}{13\!\cdots\!26}$, $\frac{220845314502837}{13\!\cdots\!26}a^{11}+\frac{502867756933344}{68\!\cdots\!13}a^{10}-\frac{43\!\cdots\!19}{27\!\cdots\!52}a^{9}-\frac{79\!\cdots\!85}{13\!\cdots\!26}a^{8}+\frac{77\!\cdots\!23}{13\!\cdots\!26}a^{7}+\frac{42\!\cdots\!73}{27\!\cdots\!52}a^{6}-\frac{23\!\cdots\!61}{27\!\cdots\!52}a^{5}-\frac{47\!\cdots\!33}{27\!\cdots\!52}a^{4}+\frac{12\!\cdots\!11}{27\!\cdots\!52}a^{3}+\frac{10\!\cdots\!33}{13\!\cdots\!26}a^{2}-\frac{90\!\cdots\!67}{27\!\cdots\!52}a-\frac{20\!\cdots\!01}{27\!\cdots\!52}$, $\frac{912084889840607}{27\!\cdots\!52}a^{11}+\frac{351039567217757}{13\!\cdots\!26}a^{10}-\frac{35\!\cdots\!85}{13\!\cdots\!26}a^{9}-\frac{49\!\cdots\!29}{27\!\cdots\!52}a^{8}+\frac{18\!\cdots\!03}{27\!\cdots\!52}a^{7}+\frac{10\!\cdots\!39}{27\!\cdots\!52}a^{6}-\frac{19\!\cdots\!11}{27\!\cdots\!52}a^{5}-\frac{39\!\cdots\!89}{13\!\cdots\!26}a^{4}+\frac{77\!\cdots\!59}{27\!\cdots\!52}a^{3}+\frac{13\!\cdots\!01}{27\!\cdots\!52}a^{2}-\frac{33\!\cdots\!85}{68\!\cdots\!13}a-\frac{10\!\cdots\!97}{13\!\cdots\!26}$, $\frac{625278639363092}{68\!\cdots\!13}a^{11}+\frac{90\!\cdots\!11}{27\!\cdots\!52}a^{10}-\frac{51\!\cdots\!15}{68\!\cdots\!13}a^{9}-\frac{36\!\cdots\!15}{13\!\cdots\!26}a^{8}+\frac{58\!\cdots\!21}{27\!\cdots\!52}a^{7}+\frac{20\!\cdots\!93}{27\!\cdots\!52}a^{6}-\frac{68\!\cdots\!37}{27\!\cdots\!52}a^{5}-\frac{22\!\cdots\!95}{27\!\cdots\!52}a^{4}+\frac{14\!\cdots\!29}{13\!\cdots\!26}a^{3}+\frac{81\!\cdots\!77}{27\!\cdots\!52}a^{2}-\frac{15\!\cdots\!83}{27\!\cdots\!52}a-\frac{17\!\cdots\!49}{13\!\cdots\!26}$, $\frac{645966781149115}{27\!\cdots\!52}a^{11}+\frac{62\!\cdots\!33}{27\!\cdots\!52}a^{10}-\frac{64\!\cdots\!21}{27\!\cdots\!52}a^{9}-\frac{48\!\cdots\!37}{27\!\cdots\!52}a^{8}+\frac{12\!\cdots\!95}{13\!\cdots\!26}a^{7}+\frac{12\!\cdots\!77}{27\!\cdots\!52}a^{6}-\frac{40\!\cdots\!49}{27\!\cdots\!52}a^{5}-\frac{32\!\cdots\!22}{68\!\cdots\!13}a^{4}+\frac{70\!\cdots\!39}{68\!\cdots\!13}a^{3}+\frac{11\!\cdots\!49}{68\!\cdots\!13}a^{2}-\frac{30\!\cdots\!89}{13\!\cdots\!26}a-\frac{86\!\cdots\!45}{27\!\cdots\!52}$
| 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: | \( 7899059.36991 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 7899059.36991 \cdot 2}{2\cdot\sqrt{24420144017624194286937}}\cr\approx \mathstrut & 0.207043316415 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 100*x^10 + 96*x^9 + 3746*x^8 - 3362*x^7 - 65339*x^6 + 51584*x^5 + 525352*x^4 - 322117*x^3 - 1551355*x^2 + 579413*x + 104959)
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^12 - x^11 - 100*x^10 + 96*x^9 + 3746*x^8 - 3362*x^7 - 65339*x^6 + 51584*x^5 + 525352*x^4 - 322117*x^3 - 1551355*x^2 + 579413*x + 104959, 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^12 - x^11 - 100*x^10 + 96*x^9 + 3746*x^8 - 3362*x^7 - 65339*x^6 + 51584*x^5 + 525352*x^4 - 322117*x^3 - 1551355*x^2 + 579413*x + 104959);
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^12 - x^11 - 100*x^10 + 96*x^9 + 3746*x^8 - 3362*x^7 - 65339*x^6 + 51584*x^5 + 525352*x^4 - 322117*x^3 - 1551355*x^2 + 579413*x + 104959);
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 cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.2166633.1, 6.6.11796113.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.12.6.1 | $x^{12} + 18 x^{8} + 81 x^{4} - 486 x^{2} + 1458$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.12.10.5 | $x^{12} - 154 x^{6} - 1421$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |