Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 52*x^10 + 112*x^9 + 1234*x^8 + 1884*x^7 - 6287*x^6 - 70*x^5 + 136036*x^4 - 166513*x^3 + 131837*x^2 - 272169*x + 1944923)
gp: K = bnfinit(y^12 - y^11 + 52*y^10 + 112*y^9 + 1234*y^8 + 1884*y^7 - 6287*y^6 - 70*y^5 + 136036*y^4 - 166513*y^3 + 131837*y^2 - 272169*y + 1944923, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 52*x^10 + 112*x^9 + 1234*x^8 + 1884*x^7 - 6287*x^6 - 70*x^5 + 136036*x^4 - 166513*x^3 + 131837*x^2 - 272169*x + 1944923);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 52*x^10 + 112*x^9 + 1234*x^8 + 1884*x^7 - 6287*x^6 - 70*x^5 + 136036*x^4 - 166513*x^3 + 131837*x^2 - 272169*x + 1944923)
\( x^{12} - x^{11} + 52 x^{10} + 112 x^{9} + 1234 x^{8} + 1884 x^{7} - 6287 x^{6} - 70 x^{5} + \cdots + 1944923 \)
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: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(727070128175028950803097\)
\(\medspace = 17^{9}\cdot 19^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(97.38\) | 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: | $17^{3/4}19^{5/6}\approx 97.37886130268147$ | ||
| Ramified primes: |
\(17\), \(19\)
| 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: | \(323=17\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{323}(1,·)$, $\chi_{323}(259,·)$, $\chi_{323}(132,·)$, $\chi_{323}(293,·)$, $\chi_{323}(273,·)$, $\chi_{323}(239,·)$, $\chi_{323}(208,·)$, $\chi_{323}(305,·)$, $\chi_{323}(183,·)$, $\chi_{323}(217,·)$, $\chi_{323}(220,·)$, $\chi_{323}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.1773593.2$^{2}$, 12.0.727070128175028950803097.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{956}a^{10}-\frac{59}{478}a^{9}+\frac{65}{478}a^{8}-\frac{79}{956}a^{7}+\frac{173}{956}a^{6}-\frac{263}{956}a^{5}-\frac{373}{956}a^{4}-\frac{9}{239}a^{3}-\frac{315}{956}a^{2}-\frac{403}{956}a+\frac{35}{478}$, $\frac{1}{75\!\cdots\!28}a^{11}+\frac{14\!\cdots\!87}{37\!\cdots\!14}a^{10}-\frac{23\!\cdots\!23}{37\!\cdots\!14}a^{9}+\frac{62\!\cdots\!21}{75\!\cdots\!28}a^{8}-\frac{99\!\cdots\!57}{75\!\cdots\!28}a^{7}-\frac{17\!\cdots\!29}{75\!\cdots\!28}a^{6}+\frac{52\!\cdots\!69}{75\!\cdots\!28}a^{5}+\frac{13\!\cdots\!95}{37\!\cdots\!14}a^{4}+\frac{24\!\cdots\!87}{75\!\cdots\!28}a^{3}-\frac{24\!\cdots\!77}{75\!\cdots\!28}a^{2}+\frac{28\!\cdots\!98}{18\!\cdots\!57}a+\frac{38\!\cdots\!41}{18\!\cdots\!57}$
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
$C_{10}\times C_{130}$, which has order $1300$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $1300$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $5$ | 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: |
$\frac{10\!\cdots\!73}{53\!\cdots\!69}a^{11}+\frac{27\!\cdots\!63}{10\!\cdots\!38}a^{10}+\frac{20\!\cdots\!63}{21\!\cdots\!76}a^{9}+\frac{23\!\cdots\!57}{53\!\cdots\!69}a^{8}+\frac{15\!\cdots\!31}{53\!\cdots\!69}a^{7}+\frac{20\!\cdots\!97}{21\!\cdots\!76}a^{6}-\frac{40\!\cdots\!37}{21\!\cdots\!76}a^{5}-\frac{57\!\cdots\!81}{21\!\cdots\!76}a^{4}+\frac{38\!\cdots\!75}{21\!\cdots\!76}a^{3}-\frac{23\!\cdots\!88}{53\!\cdots\!69}a^{2}-\frac{35\!\cdots\!15}{21\!\cdots\!76}a-\frac{27\!\cdots\!31}{21\!\cdots\!76}$, $\frac{11\!\cdots\!57}{21\!\cdots\!76}a^{11}+\frac{35\!\cdots\!18}{53\!\cdots\!69}a^{10}+\frac{30\!\cdots\!63}{10\!\cdots\!38}a^{9}+\frac{27\!\cdots\!73}{21\!\cdots\!76}a^{8}+\frac{17\!\cdots\!07}{21\!\cdots\!76}a^{7}+\frac{58\!\cdots\!47}{21\!\cdots\!76}a^{6}-\frac{10\!\cdots\!87}{21\!\cdots\!76}a^{5}-\frac{82\!\cdots\!97}{10\!\cdots\!38}a^{4}+\frac{10\!\cdots\!25}{21\!\cdots\!76}a^{3}-\frac{27\!\cdots\!01}{21\!\cdots\!76}a^{2}-\frac{25\!\cdots\!89}{53\!\cdots\!69}a+\frac{45\!\cdots\!91}{53\!\cdots\!69}$, $\frac{21\!\cdots\!19}{75\!\cdots\!28}a^{11}+\frac{39\!\cdots\!91}{75\!\cdots\!28}a^{10}+\frac{57\!\cdots\!89}{37\!\cdots\!14}a^{9}+\frac{54\!\cdots\!85}{75\!\cdots\!28}a^{8}+\frac{95\!\cdots\!86}{18\!\cdots\!57}a^{7}+\frac{62\!\cdots\!09}{37\!\cdots\!14}a^{6}+\frac{17\!\cdots\!82}{18\!\cdots\!57}a^{5}-\frac{37\!\cdots\!39}{75\!\cdots\!28}a^{4}+\frac{11\!\cdots\!25}{75\!\cdots\!28}a^{3}+\frac{19\!\cdots\!27}{18\!\cdots\!57}a^{2}+\frac{22\!\cdots\!17}{75\!\cdots\!28}a-\frac{24\!\cdots\!73}{18\!\cdots\!57}$, $\frac{28\!\cdots\!16}{18\!\cdots\!57}a^{11}+\frac{25\!\cdots\!29}{75\!\cdots\!28}a^{10}+\frac{63\!\cdots\!17}{75\!\cdots\!28}a^{9}+\frac{15\!\cdots\!35}{37\!\cdots\!14}a^{8}+\frac{22\!\cdots\!43}{75\!\cdots\!28}a^{7}+\frac{18\!\cdots\!67}{18\!\cdots\!57}a^{6}+\frac{39\!\cdots\!39}{37\!\cdots\!14}a^{5}-\frac{58\!\cdots\!82}{18\!\cdots\!57}a^{4}+\frac{19\!\cdots\!75}{75\!\cdots\!28}a^{3}+\frac{10\!\cdots\!11}{75\!\cdots\!28}a^{2}+\frac{77\!\cdots\!36}{18\!\cdots\!57}a+\frac{62\!\cdots\!43}{75\!\cdots\!28}$, $\frac{16135718640171}{83\!\cdots\!58}a^{11}+\frac{44808272943761}{83\!\cdots\!58}a^{10}+\frac{950536498344083}{83\!\cdots\!58}a^{9}+\frac{46\!\cdots\!97}{83\!\cdots\!58}a^{8}+\frac{17\!\cdots\!42}{41\!\cdots\!29}a^{7}+\frac{11\!\cdots\!69}{83\!\cdots\!58}a^{6}+\frac{18\!\cdots\!33}{83\!\cdots\!58}a^{5}-\frac{19\!\cdots\!31}{41\!\cdots\!29}a^{4}-\frac{20\!\cdots\!25}{41\!\cdots\!29}a^{3}+\frac{12\!\cdots\!54}{41\!\cdots\!29}a^{2}+\frac{38\!\cdots\!82}{41\!\cdots\!29}a+\frac{81\!\cdots\!05}{83\!\cdots\!58}$
| 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: | \( 5500.24745416 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 5500.24745416 \cdot 1300}{2\cdot\sqrt{727070128175028950803097}}\cr\approx \mathstrut & 0.257980360032 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 52*x^10 + 112*x^9 + 1234*x^8 + 1884*x^7 - 6287*x^6 - 70*x^5 + 136036*x^4 - 166513*x^3 + 131837*x^2 - 272169*x + 1944923)
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 + 52*x^10 + 112*x^9 + 1234*x^8 + 1884*x^7 - 6287*x^6 - 70*x^5 + 136036*x^4 - 166513*x^3 + 131837*x^2 - 272169*x + 1944923, 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 + 52*x^10 + 112*x^9 + 1234*x^8 + 1884*x^7 - 6287*x^6 - 70*x^5 + 136036*x^4 - 166513*x^3 + 131837*x^2 - 272169*x + 1944923);
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 + 52*x^10 + 112*x^9 + 1234*x^8 + 1884*x^7 - 6287*x^6 - 70*x^5 + 136036*x^4 - 166513*x^3 + 131837*x^2 - 272169*x + 1944923);
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
| \(\Q(\sqrt{17}) \), 3.3.361.1, 4.0.1773593.2, 6.6.640267073.1 |
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 | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(19\)
| 19.6.5.2 | $x^{6} + 152$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 19.6.5.2 | $x^{6} + 152$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |