Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 270*x^10 - 275*x^9 + 20801*x^8 - 24235*x^7 + 615964*x^6 - 658945*x^5 + 8871411*x^4 - 3368575*x^3 + 65405755*x^2 + 15856409*x + 249089881)
gp: K = bnfinit(y^12 - y^11 + 270*y^10 - 275*y^9 + 20801*y^8 - 24235*y^7 + 615964*y^6 - 658945*y^5 + 8871411*y^4 - 3368575*y^3 + 65405755*y^2 + 15856409*y + 249089881, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 270*x^10 - 275*x^9 + 20801*x^8 - 24235*x^7 + 615964*x^6 - 658945*x^5 + 8871411*x^4 - 3368575*x^3 + 65405755*x^2 + 15856409*x + 249089881);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 270*x^10 - 275*x^9 + 20801*x^8 - 24235*x^7 + 615964*x^6 - 658945*x^5 + 8871411*x^4 - 3368575*x^3 + 65405755*x^2 + 15856409*x + 249089881)
\( x^{12} - x^{11} + 270 x^{10} - 275 x^{9} + 20801 x^{8} - 24235 x^{7} + 615964 x^{6} - 658945 x^{5} + \cdots + 249089881 \)
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: |
\(6499157928205420080078125\)
\(\medspace = 5^{9}\cdot 13^{10}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(116.88\) | 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: | $5^{3/4}13^{5/6}17^{1/2}\approx 116.87943159008147$ | ||
| Ramified primes: |
\(5\), \(13\), \(17\)
| 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: | \(1105=5\cdot 13\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1105}(256,·)$, $\chi_{1105}(1,·)$, $\chi_{1105}(322,·)$, $\chi_{1105}(407,·)$, $\chi_{1105}(1004,·)$, $\chi_{1105}(883,·)$, $\chi_{1105}(628,·)$, $\chi_{1105}(341,·)$, $\chi_{1105}(662,·)$, $\chi_{1105}(919,·)$, $\chi_{1105}(664,·)$, $\chi_{1105}(543,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.6105125.1$^{2}$, 12.0.6499157928205420080078125.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{31}a^{9}-\frac{12}{31}a^{8}-\frac{6}{31}a^{7}-\frac{3}{31}a^{6}-\frac{15}{31}a^{5}-\frac{7}{31}a^{4}-\frac{13}{31}a^{3}-\frac{6}{31}a^{2}-\frac{6}{31}a+\frac{5}{31}$, $\frac{1}{31}a^{10}+\frac{5}{31}a^{8}-\frac{13}{31}a^{7}+\frac{11}{31}a^{6}-\frac{1}{31}a^{5}-\frac{4}{31}a^{4}-\frac{7}{31}a^{3}+\frac{15}{31}a^{2}-\frac{5}{31}a-\frac{2}{31}$, $\frac{1}{31\!\cdots\!41}a^{11}+\frac{17\!\cdots\!83}{31\!\cdots\!41}a^{10}+\frac{25\!\cdots\!46}{31\!\cdots\!41}a^{9}+\frac{17\!\cdots\!48}{31\!\cdots\!41}a^{8}+\frac{32\!\cdots\!38}{10\!\cdots\!11}a^{7}+\frac{12\!\cdots\!05}{31\!\cdots\!41}a^{6}+\frac{67\!\cdots\!26}{31\!\cdots\!41}a^{5}-\frac{34\!\cdots\!82}{31\!\cdots\!41}a^{4}+\frac{73\!\cdots\!42}{31\!\cdots\!41}a^{3}-\frac{96\!\cdots\!26}{31\!\cdots\!41}a^{2}+\frac{13\!\cdots\!43}{31\!\cdots\!41}a-\frac{20\!\cdots\!73}{31\!\cdots\!41}$
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
$C_{2}\times C_{2}\times C_{2}\times C_{5746}$, which has order $45968$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $45968$ (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: |
$\frac{11\!\cdots\!71}{47\!\cdots\!11}a^{11}-\frac{34\!\cdots\!07}{47\!\cdots\!11}a^{10}+\frac{97\!\cdots\!55}{15\!\cdots\!81}a^{9}-\frac{94\!\cdots\!97}{47\!\cdots\!11}a^{8}+\frac{21\!\cdots\!23}{47\!\cdots\!11}a^{7}-\frac{73\!\cdots\!90}{47\!\cdots\!11}a^{6}+\frac{55\!\cdots\!42}{47\!\cdots\!11}a^{5}-\frac{18\!\cdots\!03}{47\!\cdots\!11}a^{4}+\frac{60\!\cdots\!75}{47\!\cdots\!11}a^{3}-\frac{10\!\cdots\!59}{47\!\cdots\!11}a^{2}+\frac{18\!\cdots\!47}{47\!\cdots\!11}a-\frac{47\!\cdots\!24}{47\!\cdots\!11}$, $\frac{17\!\cdots\!10}{96\!\cdots\!81}a^{11}-\frac{52\!\cdots\!25}{96\!\cdots\!81}a^{10}+\frac{40\!\cdots\!10}{96\!\cdots\!81}a^{9}-\frac{13\!\cdots\!55}{96\!\cdots\!81}a^{8}+\frac{20\!\cdots\!65}{96\!\cdots\!81}a^{7}-\frac{10\!\cdots\!80}{96\!\cdots\!81}a^{6}+\frac{32\!\cdots\!94}{31\!\cdots\!51}a^{5}-\frac{23\!\cdots\!90}{96\!\cdots\!81}a^{4}-\frac{27\!\cdots\!80}{96\!\cdots\!81}a^{3}-\frac{22\!\cdots\!60}{96\!\cdots\!81}a^{2}-\frac{21\!\cdots\!40}{96\!\cdots\!81}a-\frac{10\!\cdots\!84}{96\!\cdots\!81}$, $\frac{49\!\cdots\!80}{96\!\cdots\!81}a^{11}-\frac{59\!\cdots\!69}{31\!\cdots\!51}a^{10}+\frac{11\!\cdots\!30}{96\!\cdots\!81}a^{9}-\frac{46\!\cdots\!90}{96\!\cdots\!81}a^{8}+\frac{59\!\cdots\!20}{96\!\cdots\!81}a^{7}-\frac{31\!\cdots\!15}{96\!\cdots\!81}a^{6}+\frac{33\!\cdots\!51}{96\!\cdots\!81}a^{5}-\frac{69\!\cdots\!95}{96\!\cdots\!81}a^{4}-\frac{70\!\cdots\!40}{96\!\cdots\!81}a^{3}-\frac{65\!\cdots\!30}{96\!\cdots\!81}a^{2}-\frac{59\!\cdots\!95}{96\!\cdots\!81}a+\frac{71\!\cdots\!12}{96\!\cdots\!81}$, $\frac{24\!\cdots\!56}{31\!\cdots\!41}a^{11}-\frac{39\!\cdots\!74}{31\!\cdots\!41}a^{10}+\frac{64\!\cdots\!19}{31\!\cdots\!41}a^{9}-\frac{33\!\cdots\!78}{10\!\cdots\!11}a^{8}+\frac{44\!\cdots\!02}{31\!\cdots\!41}a^{7}-\frac{80\!\cdots\!18}{31\!\cdots\!41}a^{6}+\frac{10\!\cdots\!46}{31\!\cdots\!41}a^{5}-\frac{16\!\cdots\!94}{31\!\cdots\!41}a^{4}+\frac{12\!\cdots\!30}{31\!\cdots\!41}a^{3}-\frac{39\!\cdots\!91}{31\!\cdots\!41}a^{2}+\frac{47\!\cdots\!24}{31\!\cdots\!41}a-\frac{66\!\cdots\!81}{31\!\cdots\!41}$, $\frac{12\!\cdots\!61}{31\!\cdots\!41}a^{11}-\frac{15\!\cdots\!99}{31\!\cdots\!41}a^{10}+\frac{34\!\cdots\!39}{31\!\cdots\!41}a^{9}-\frac{39\!\cdots\!04}{31\!\cdots\!41}a^{8}+\frac{26\!\cdots\!15}{31\!\cdots\!41}a^{7}-\frac{29\!\cdots\!91}{31\!\cdots\!41}a^{6}+\frac{73\!\cdots\!99}{31\!\cdots\!41}a^{5}-\frac{51\!\cdots\!65}{31\!\cdots\!41}a^{4}+\frac{93\!\cdots\!54}{31\!\cdots\!41}a^{3}+\frac{35\!\cdots\!32}{31\!\cdots\!41}a^{2}+\frac{39\!\cdots\!97}{31\!\cdots\!41}a+\frac{75\!\cdots\!29}{31\!\cdots\!41}$
| 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: | \( 615.54450504 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 615.54450504 \cdot 45968}{2\cdot\sqrt{6499157928205420080078125}}\cr\approx \mathstrut & 0.34145677696 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 270*x^10 - 275*x^9 + 20801*x^8 - 24235*x^7 + 615964*x^6 - 658945*x^5 + 8871411*x^4 - 3368575*x^3 + 65405755*x^2 + 15856409*x + 249089881)
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 + 270*x^10 - 275*x^9 + 20801*x^8 - 24235*x^7 + 615964*x^6 - 658945*x^5 + 8871411*x^4 - 3368575*x^3 + 65405755*x^2 + 15856409*x + 249089881, 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 + 270*x^10 - 275*x^9 + 20801*x^8 - 24235*x^7 + 615964*x^6 - 658945*x^5 + 8871411*x^4 - 3368575*x^3 + 65405755*x^2 + 15856409*x + 249089881);
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 + 270*x^10 - 275*x^9 + 20801*x^8 - 24235*x^7 + 615964*x^6 - 658945*x^5 + 8871411*x^4 - 3368575*x^3 + 65405755*x^2 + 15856409*x + 249089881);
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
| \(\Q(\sqrt{5}) \), 3.3.169.1, 4.0.6105125.1, 6.6.3570125.1 |
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 | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.1.0.1}{1} }^{12}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(13\)
| 13.12.10.5 | $x^{12} - 1586 x^{6} - 198575$ | $6$ | $2$ | $10$ | $C_{12}$ | $[\ ]_{6}^{2}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |