Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 23*x^10 - 241*x^9 + 963*x^8 - 5332*x^7 + 35774*x^6 - 128756*x^5 + 492276*x^4 - 1873692*x^3 + 5096688*x^2 - 7454263*x + 7283051)
gp: K = bnfinit(y^12 - y^11 + 23*y^10 - 241*y^9 + 963*y^8 - 5332*y^7 + 35774*y^6 - 128756*y^5 + 492276*y^4 - 1873692*y^3 + 5096688*y^2 - 7454263*y + 7283051, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 23*x^10 - 241*x^9 + 963*x^8 - 5332*x^7 + 35774*x^6 - 128756*x^5 + 492276*x^4 - 1873692*x^3 + 5096688*x^2 - 7454263*x + 7283051);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 23*x^10 - 241*x^9 + 963*x^8 - 5332*x^7 + 35774*x^6 - 128756*x^5 + 492276*x^4 - 1873692*x^3 + 5096688*x^2 - 7454263*x + 7283051)
\( x^{12} - x^{11} + 23 x^{10} - 241 x^{9} + 963 x^{8} - 5332 x^{7} + 35774 x^{6} - 128756 x^{5} + \cdots + 7283051 \)
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: |
\(1335225603550355658203125\)
\(\medspace = 5^{9}\cdot 7^{8}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(102.44\) | 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: | $5^{3/4}7^{2/3}17^{3/4}\approx 102.4384238436466$ | ||
| 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: | \(\Q(\sqrt{85}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | 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$$\chi_{595}(256,·)$, $\chi_{595}(1,·)$, $\chi_{595}(86,·)$, $\chi_{595}(582,·)$, $\chi_{595}(72,·)$, $\chi_{595}(169,·)$, $\chi_{595}(268,·)$, $\chi_{595}(424,·)$, $\chi_{595}(242,·)$, $\chi_{595}(438,·)$, $\chi_{595}(183,·)$, $\chi_{595}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.614125.2$^{2}$, 12.0.1335225603550355658203125.2$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{57}a^{9}-\frac{2}{19}a^{8}+\frac{28}{57}a^{7}-\frac{4}{57}a^{6}+\frac{4}{57}a^{5}+\frac{7}{19}a^{4}+\frac{5}{57}a^{3}+\frac{23}{57}a^{2}-\frac{3}{19}a-\frac{10}{57}$, $\frac{1}{57}a^{10}-\frac{8}{57}a^{8}-\frac{7}{57}a^{7}-\frac{20}{57}a^{6}-\frac{4}{19}a^{5}+\frac{17}{57}a^{4}-\frac{4}{57}a^{3}+\frac{5}{19}a^{2}-\frac{7}{57}a-\frac{1}{19}$, $\frac{1}{12\!\cdots\!87}a^{11}-\frac{13\!\cdots\!84}{41\!\cdots\!29}a^{10}-\frac{28\!\cdots\!96}{64\!\cdots\!73}a^{9}-\frac{33\!\cdots\!29}{12\!\cdots\!87}a^{8}-\frac{69\!\cdots\!69}{12\!\cdots\!87}a^{7}+\frac{11\!\cdots\!76}{41\!\cdots\!29}a^{6}+\frac{62\!\cdots\!53}{12\!\cdots\!87}a^{5}-\frac{90\!\cdots\!96}{12\!\cdots\!87}a^{4}+\frac{18\!\cdots\!36}{41\!\cdots\!29}a^{3}+\frac{42\!\cdots\!77}{12\!\cdots\!87}a^{2}+\frac{12\!\cdots\!25}{41\!\cdots\!29}a-\frac{78\!\cdots\!19}{41\!\cdots\!29}$
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
$C_{1492}$, which has order $1492$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $746$ (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: |
$\frac{85\!\cdots\!15}{21\!\cdots\!91}a^{11}-\frac{69\!\cdots\!40}{21\!\cdots\!91}a^{10}+\frac{14\!\cdots\!70}{21\!\cdots\!91}a^{9}-\frac{21\!\cdots\!95}{21\!\cdots\!91}a^{8}+\frac{53\!\cdots\!80}{21\!\cdots\!91}a^{7}-\frac{41\!\cdots\!70}{21\!\cdots\!91}a^{6}+\frac{25\!\cdots\!39}{21\!\cdots\!91}a^{5}-\frac{71\!\cdots\!65}{21\!\cdots\!91}a^{4}+\frac{30\!\cdots\!15}{21\!\cdots\!91}a^{3}-\frac{11\!\cdots\!35}{21\!\cdots\!91}a^{2}+\frac{18\!\cdots\!30}{21\!\cdots\!91}a-\frac{34\!\cdots\!44}{21\!\cdots\!91}$, $\frac{85\!\cdots\!15}{21\!\cdots\!91}a^{11}-\frac{69\!\cdots\!40}{21\!\cdots\!91}a^{10}+\frac{14\!\cdots\!70}{21\!\cdots\!91}a^{9}-\frac{21\!\cdots\!95}{21\!\cdots\!91}a^{8}+\frac{53\!\cdots\!80}{21\!\cdots\!91}a^{7}-\frac{41\!\cdots\!70}{21\!\cdots\!91}a^{6}+\frac{25\!\cdots\!39}{21\!\cdots\!91}a^{5}-\frac{71\!\cdots\!65}{21\!\cdots\!91}a^{4}+\frac{30\!\cdots\!15}{21\!\cdots\!91}a^{3}-\frac{11\!\cdots\!35}{21\!\cdots\!91}a^{2}+\frac{18\!\cdots\!30}{21\!\cdots\!91}a-\frac{12\!\cdots\!53}{21\!\cdots\!91}$, $\frac{38182842187471}{26\!\cdots\!99}a^{11}+\frac{189947591477102}{26\!\cdots\!99}a^{10}-\frac{234558916958142}{26\!\cdots\!99}a^{9}-\frac{42\!\cdots\!09}{26\!\cdots\!99}a^{8}-\frac{29\!\cdots\!38}{26\!\cdots\!99}a^{7}+\frac{21\!\cdots\!72}{26\!\cdots\!99}a^{6}-\frac{16\!\cdots\!33}{14\!\cdots\!21}a^{5}+\frac{49\!\cdots\!82}{26\!\cdots\!99}a^{4}-\frac{34\!\cdots\!85}{26\!\cdots\!99}a^{3}+\frac{10\!\cdots\!75}{26\!\cdots\!99}a^{2}-\frac{16\!\cdots\!21}{26\!\cdots\!99}a+\frac{18\!\cdots\!39}{26\!\cdots\!99}$, $\frac{67\!\cdots\!44}{41\!\cdots\!29}a^{11}+\frac{16\!\cdots\!58}{41\!\cdots\!29}a^{10}+\frac{18\!\cdots\!02}{41\!\cdots\!29}a^{9}-\frac{13\!\cdots\!90}{41\!\cdots\!29}a^{8}+\frac{26\!\cdots\!23}{41\!\cdots\!29}a^{7}-\frac{24\!\cdots\!61}{41\!\cdots\!29}a^{6}+\frac{19\!\cdots\!08}{41\!\cdots\!29}a^{5}-\frac{42\!\cdots\!83}{41\!\cdots\!29}a^{4}+\frac{22\!\cdots\!77}{41\!\cdots\!29}a^{3}-\frac{93\!\cdots\!25}{41\!\cdots\!29}a^{2}+\frac{15\!\cdots\!69}{41\!\cdots\!29}a+\frac{80\!\cdots\!59}{41\!\cdots\!29}$, $\frac{25\!\cdots\!23}{41\!\cdots\!29}a^{11}+\frac{21\!\cdots\!43}{41\!\cdots\!29}a^{10}+\frac{53\!\cdots\!40}{41\!\cdots\!29}a^{9}-\frac{50\!\cdots\!10}{41\!\cdots\!29}a^{8}+\frac{12\!\cdots\!54}{41\!\cdots\!29}a^{7}-\frac{92\!\cdots\!17}{41\!\cdots\!29}a^{6}+\frac{67\!\cdots\!14}{41\!\cdots\!29}a^{5}-\frac{15\!\cdots\!01}{41\!\cdots\!29}a^{4}+\frac{66\!\cdots\!13}{41\!\cdots\!29}a^{3}-\frac{26\!\cdots\!57}{41\!\cdots\!29}a^{2}+\frac{43\!\cdots\!10}{41\!\cdots\!29}a+\frac{25\!\cdots\!32}{41\!\cdots\!29}$
| 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: | \( 3699.65687174 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3699.65687174 \cdot 1492}{2\cdot\sqrt{1335225603550355658203125}}\cr\approx \mathstrut & 0.146961020387 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 23*x^10 - 241*x^9 + 963*x^8 - 5332*x^7 + 35774*x^6 - 128756*x^5 + 492276*x^4 - 1873692*x^3 + 5096688*x^2 - 7454263*x + 7283051)
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 + 23*x^10 - 241*x^9 + 963*x^8 - 5332*x^7 + 35774*x^6 - 128756*x^5 + 492276*x^4 - 1873692*x^3 + 5096688*x^2 - 7454263*x + 7283051, 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 + 23*x^10 - 241*x^9 + 963*x^8 - 5332*x^7 + 35774*x^6 - 128756*x^5 + 492276*x^4 - 1873692*x^3 + 5096688*x^2 - 7454263*x + 7283051);
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 + 23*x^10 - 241*x^9 + 963*x^8 - 5332*x^7 + 35774*x^6 - 128756*x^5 + 492276*x^4 - 1873692*x^3 + 5096688*x^2 - 7454263*x + 7283051);
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
| \(\Q(\sqrt{85}) \), \(\Q(\zeta_{7})^+\), 4.0.614125.2, 6.6.1474514125.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.9.3 | $x^{12} + 75 x^{4} - 375$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(7\)
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(17\)
| 17.12.9.3 | $x^{12} + 289 x^{4} - 68782$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |