Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 152*x^10 + 151*x^9 + 8594*x^8 - 8443*x^7 - 228593*x^6 + 234291*x^5 + 2932130*x^4 - 3275066*x^3 - 15786160*x^2 + 17913708*x + 16784881)
gp: K = bnfinit(y^12 - y^11 - 152*y^10 + 151*y^9 + 8594*y^8 - 8443*y^7 - 228593*y^6 + 234291*y^5 + 2932130*y^4 - 3275066*y^3 - 15786160*y^2 + 17913708*y + 16784881, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 152*x^10 + 151*x^9 + 8594*x^8 - 8443*x^7 - 228593*x^6 + 234291*x^5 + 2932130*x^4 - 3275066*x^3 - 15786160*x^2 + 17913708*x + 16784881);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 152*x^10 + 151*x^9 + 8594*x^8 - 8443*x^7 - 228593*x^6 + 234291*x^5 + 2932130*x^4 - 3275066*x^3 - 15786160*x^2 + 17913708*x + 16784881)
\( x^{12} - x^{11} - 152 x^{10} + 151 x^{9} + 8594 x^{8} - 8443 x^{7} - 228593 x^{6} + 234291 x^{5} + \cdots + 16784881 \)
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: |
\(13316925417050158203125\)
\(\medspace = 5^{9}\cdot 7^{10}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(69.78\) | 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: | $5^{3/4}7^{5/6}17^{1/2}\approx 69.77507799528695$ | ||
| Ramified primes: |
\(5\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(595=5\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{595}(256,·)$, $\chi_{595}(1,·)$, $\chi_{595}(324,·)$, $\chi_{595}(86,·)$, $\chi_{595}(33,·)$, $\chi_{595}(458,·)$, $\chi_{595}(237,·)$, $\chi_{595}(494,·)$, $\chi_{595}(239,·)$, $\chi_{595}(577,·)$, $\chi_{595}(152,·)$, $\chi_{595}(118,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{41}a^{9}-\frac{19}{41}a^{8}+\frac{18}{41}a^{7}-\frac{11}{41}a^{6}+\frac{12}{41}a^{5}+\frac{14}{41}a^{4}+\frac{16}{41}a^{3}-\frac{17}{41}a^{2}+\frac{5}{41}a+\frac{15}{41}$, $\frac{1}{17027669}a^{10}+\frac{20740}{17027669}a^{9}+\frac{3790959}{17027669}a^{8}+\frac{6469326}{17027669}a^{7}+\frac{4775057}{17027669}a^{6}-\frac{7618901}{17027669}a^{5}-\frac{934848}{17027669}a^{4}-\frac{4229574}{17027669}a^{3}-\frac{3680827}{17027669}a^{2}+\frac{425373}{17027669}a-\frac{214600}{587161}$, $\frac{1}{19\!\cdots\!19}a^{11}-\frac{458042871513585}{19\!\cdots\!19}a^{10}+\frac{32\!\cdots\!98}{19\!\cdots\!19}a^{9}+\frac{45\!\cdots\!84}{19\!\cdots\!19}a^{8}+\frac{64\!\cdots\!76}{68\!\cdots\!11}a^{7}+\frac{88\!\cdots\!16}{19\!\cdots\!19}a^{6}+\frac{33\!\cdots\!03}{19\!\cdots\!19}a^{5}+\frac{49\!\cdots\!66}{19\!\cdots\!19}a^{4}+\frac{67\!\cdots\!10}{19\!\cdots\!19}a^{3}-\frac{34\!\cdots\!26}{19\!\cdots\!19}a^{2}+\frac{95\!\cdots\!27}{19\!\cdots\!19}a-\frac{302094311419534}{11\!\cdots\!99}$
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 $2$ (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: |
$\frac{13528093523532}{13\!\cdots\!39}a^{11}+\frac{31293249118572}{13\!\cdots\!39}a^{10}-\frac{19\!\cdots\!92}{13\!\cdots\!39}a^{9}-\frac{44\!\cdots\!68}{13\!\cdots\!39}a^{8}+\frac{10\!\cdots\!41}{13\!\cdots\!39}a^{7}+\frac{23\!\cdots\!28}{13\!\cdots\!39}a^{6}-\frac{26\!\cdots\!71}{13\!\cdots\!39}a^{5}-\frac{50\!\cdots\!80}{13\!\cdots\!39}a^{4}+\frac{29\!\cdots\!38}{13\!\cdots\!39}a^{3}+\frac{42\!\cdots\!08}{13\!\cdots\!39}a^{2}-\frac{12\!\cdots\!03}{13\!\cdots\!39}a-\frac{5404127729140}{82773381319}$, $\frac{621334863921520}{10\!\cdots\!99}a^{11}+\frac{21\!\cdots\!80}{10\!\cdots\!99}a^{10}-\frac{94\!\cdots\!00}{10\!\cdots\!99}a^{9}-\frac{30\!\cdots\!20}{10\!\cdots\!99}a^{8}+\frac{52\!\cdots\!00}{10\!\cdots\!99}a^{7}+\frac{15\!\cdots\!60}{10\!\cdots\!99}a^{6}-\frac{13\!\cdots\!61}{10\!\cdots\!99}a^{5}-\frac{32\!\cdots\!00}{10\!\cdots\!99}a^{4}+\frac{16\!\cdots\!45}{10\!\cdots\!99}a^{3}+\frac{27\!\cdots\!00}{10\!\cdots\!99}a^{2}-\frac{73\!\cdots\!05}{10\!\cdots\!99}a-\frac{7618520912520}{159735560419}$, $\frac{819048231991960}{10\!\cdots\!99}a^{11}+\frac{272647235402579}{10\!\cdots\!99}a^{10}-\frac{11\!\cdots\!00}{10\!\cdots\!99}a^{9}-\frac{50\!\cdots\!70}{10\!\cdots\!99}a^{8}+\frac{60\!\cdots\!60}{10\!\cdots\!99}a^{7}+\frac{34\!\cdots\!65}{10\!\cdots\!99}a^{6}-\frac{13\!\cdots\!80}{10\!\cdots\!99}a^{5}-\frac{91\!\cdots\!90}{10\!\cdots\!99}a^{4}+\frac{13\!\cdots\!60}{10\!\cdots\!99}a^{3}+\frac{95\!\cdots\!95}{10\!\cdots\!99}a^{2}-\frac{50\!\cdots\!00}{10\!\cdots\!99}a-\frac{190696964755979}{6549157977179}$, $\frac{15\!\cdots\!28}{68\!\cdots\!11}a^{11}+\frac{39\!\cdots\!13}{19\!\cdots\!19}a^{10}-\frac{74\!\cdots\!32}{19\!\cdots\!19}a^{9}-\frac{55\!\cdots\!58}{19\!\cdots\!19}a^{8}+\frac{45\!\cdots\!01}{19\!\cdots\!19}a^{7}+\frac{27\!\cdots\!23}{19\!\cdots\!19}a^{6}-\frac{12\!\cdots\!11}{19\!\cdots\!19}a^{5}-\frac{55\!\cdots\!90}{19\!\cdots\!19}a^{4}+\frac{17\!\cdots\!38}{19\!\cdots\!19}a^{3}+\frac{43\!\cdots\!73}{19\!\cdots\!19}a^{2}-\frac{88\!\cdots\!63}{19\!\cdots\!19}a-\frac{42\!\cdots\!41}{11\!\cdots\!99}$, $\frac{66\!\cdots\!08}{19\!\cdots\!19}a^{11}-\frac{37\!\cdots\!33}{19\!\cdots\!19}a^{10}-\frac{96\!\cdots\!68}{19\!\cdots\!19}a^{9}-\frac{43\!\cdots\!62}{19\!\cdots\!19}a^{8}+\frac{50\!\cdots\!99}{19\!\cdots\!19}a^{7}+\frac{10\!\cdots\!37}{19\!\cdots\!19}a^{6}-\frac{11\!\cdots\!30}{19\!\cdots\!19}a^{5}-\frac{41\!\cdots\!10}{19\!\cdots\!19}a^{4}+\frac{12\!\cdots\!07}{19\!\cdots\!19}a^{3}+\frac{56\!\cdots\!27}{19\!\cdots\!19}a^{2}-\frac{44\!\cdots\!42}{19\!\cdots\!19}a-\frac{13\!\cdots\!79}{11\!\cdots\!99}$, $\frac{70\!\cdots\!76}{19\!\cdots\!19}a^{11}-\frac{39\!\cdots\!75}{19\!\cdots\!19}a^{10}-\frac{87\!\cdots\!83}{19\!\cdots\!19}a^{9}+\frac{51\!\cdots\!03}{19\!\cdots\!19}a^{8}+\frac{35\!\cdots\!71}{19\!\cdots\!19}a^{7}-\frac{22\!\cdots\!06}{19\!\cdots\!19}a^{6}-\frac{51\!\cdots\!13}{19\!\cdots\!19}a^{5}+\frac{41\!\cdots\!87}{19\!\cdots\!19}a^{4}+\frac{85\!\cdots\!46}{19\!\cdots\!19}a^{3}-\frac{28\!\cdots\!78}{19\!\cdots\!19}a^{2}+\frac{22\!\cdots\!87}{19\!\cdots\!19}a+\frac{15\!\cdots\!87}{11\!\cdots\!99}$, $\frac{56\!\cdots\!98}{19\!\cdots\!19}a^{11}+\frac{11\!\cdots\!88}{19\!\cdots\!19}a^{10}-\frac{82\!\cdots\!72}{19\!\cdots\!19}a^{9}-\frac{16\!\cdots\!65}{19\!\cdots\!19}a^{8}+\frac{44\!\cdots\!48}{19\!\cdots\!19}a^{7}+\frac{84\!\cdots\!36}{19\!\cdots\!19}a^{6}-\frac{10\!\cdots\!51}{19\!\cdots\!19}a^{5}-\frac{18\!\cdots\!58}{19\!\cdots\!19}a^{4}+\frac{12\!\cdots\!74}{19\!\cdots\!19}a^{3}+\frac{16\!\cdots\!17}{19\!\cdots\!19}a^{2}-\frac{50\!\cdots\!65}{19\!\cdots\!19}a-\frac{22\!\cdots\!24}{11\!\cdots\!99}$, $\frac{18\!\cdots\!03}{19\!\cdots\!19}a^{11}+\frac{64\!\cdots\!91}{19\!\cdots\!19}a^{10}-\frac{26\!\cdots\!42}{19\!\cdots\!19}a^{9}-\frac{89\!\cdots\!40}{19\!\cdots\!19}a^{8}+\frac{13\!\cdots\!44}{19\!\cdots\!19}a^{7}+\frac{44\!\cdots\!88}{19\!\cdots\!19}a^{6}-\frac{31\!\cdots\!67}{19\!\cdots\!19}a^{5}-\frac{89\!\cdots\!08}{19\!\cdots\!19}a^{4}+\frac{32\!\cdots\!50}{19\!\cdots\!19}a^{3}+\frac{66\!\cdots\!51}{19\!\cdots\!19}a^{2}-\frac{13\!\cdots\!12}{19\!\cdots\!19}a-\frac{63\!\cdots\!43}{11\!\cdots\!99}$, $\frac{35\!\cdots\!80}{19\!\cdots\!19}a^{11}+\frac{94\!\cdots\!47}{19\!\cdots\!19}a^{10}-\frac{52\!\cdots\!75}{19\!\cdots\!19}a^{9}-\frac{13\!\cdots\!19}{19\!\cdots\!19}a^{8}+\frac{28\!\cdots\!68}{19\!\cdots\!19}a^{7}+\frac{68\!\cdots\!26}{19\!\cdots\!19}a^{6}-\frac{72\!\cdots\!47}{19\!\cdots\!19}a^{5}-\frac{14\!\cdots\!02}{19\!\cdots\!19}a^{4}+\frac{83\!\cdots\!82}{19\!\cdots\!19}a^{3}+\frac{12\!\cdots\!67}{19\!\cdots\!19}a^{2}-\frac{35\!\cdots\!26}{19\!\cdots\!19}a-\frac{15\!\cdots\!73}{11\!\cdots\!99}$, $\frac{27\!\cdots\!12}{19\!\cdots\!19}a^{11}+\frac{53\!\cdots\!68}{19\!\cdots\!19}a^{10}-\frac{39\!\cdots\!69}{19\!\cdots\!19}a^{9}-\frac{76\!\cdots\!78}{19\!\cdots\!19}a^{8}+\frac{21\!\cdots\!54}{19\!\cdots\!19}a^{7}+\frac{13\!\cdots\!91}{68\!\cdots\!11}a^{6}-\frac{50\!\cdots\!79}{19\!\cdots\!19}a^{5}-\frac{84\!\cdots\!17}{19\!\cdots\!19}a^{4}+\frac{54\!\cdots\!20}{19\!\cdots\!19}a^{3}+\frac{70\!\cdots\!63}{19\!\cdots\!19}a^{2}-\frac{21\!\cdots\!48}{19\!\cdots\!19}a-\frac{92\!\cdots\!66}{11\!\cdots\!99}$, $\frac{16\!\cdots\!25}{19\!\cdots\!19}a^{11}-\frac{94\!\cdots\!53}{19\!\cdots\!19}a^{10}-\frac{20\!\cdots\!70}{19\!\cdots\!19}a^{9}+\frac{12\!\cdots\!45}{19\!\cdots\!19}a^{8}+\frac{82\!\cdots\!13}{19\!\cdots\!19}a^{7}-\frac{52\!\cdots\!28}{19\!\cdots\!19}a^{6}-\frac{12\!\cdots\!24}{19\!\cdots\!19}a^{5}+\frac{94\!\cdots\!30}{19\!\cdots\!19}a^{4}+\frac{23\!\cdots\!55}{19\!\cdots\!19}a^{3}-\frac{63\!\cdots\!25}{19\!\cdots\!19}a^{2}+\frac{48\!\cdots\!54}{19\!\cdots\!19}a+\frac{33\!\cdots\!67}{11\!\cdots\!99}$
| 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: | \( 6790562.47053 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 6790562.47053 \cdot 2}{2\cdot\sqrt{13316925417050158203125}}\cr\approx \mathstrut & 0.241025899973 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 152*x^10 + 151*x^9 + 8594*x^8 - 8443*x^7 - 228593*x^6 + 234291*x^5 + 2932130*x^4 - 3275066*x^3 - 15786160*x^2 + 17913708*x + 16784881)
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 - 152*x^10 + 151*x^9 + 8594*x^8 - 8443*x^7 - 228593*x^6 + 234291*x^5 + 2932130*x^4 - 3275066*x^3 - 15786160*x^2 + 17913708*x + 16784881, 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 - 152*x^10 + 151*x^9 + 8594*x^8 - 8443*x^7 - 228593*x^6 + 234291*x^5 + 2932130*x^4 - 3275066*x^3 - 15786160*x^2 + 17913708*x + 16784881);
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 - 152*x^10 + 151*x^9 + 8594*x^8 - 8443*x^7 - 228593*x^6 + 234291*x^5 + 2932130*x^4 - 3275066*x^3 - 15786160*x^2 + 17913708*x + 16784881);
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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.4.1770125.1, 6.6.300125.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.1.0.1}{1} }^{12}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.1.0.1}{1} }^{12}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(7\)
| 7.12.10.5 | $x^{12} - 154 x^{6} - 1421$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |