Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 44*x^10 + 63*x^9 + 574*x^8 - 851*x^7 - 2964*x^6 + 4032*x^5 + 6247*x^4 - 6815*x^3 - 4744*x^2 + 2450*x + 1291)
gp: K = bnfinit(y^12 - y^11 - 44*y^10 + 63*y^9 + 574*y^8 - 851*y^7 - 2964*y^6 + 4032*y^5 + 6247*y^4 - 6815*y^3 - 4744*y^2 + 2450*y + 1291, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 44*x^10 + 63*x^9 + 574*x^8 - 851*x^7 - 2964*x^6 + 4032*x^5 + 6247*x^4 - 6815*x^3 - 4744*x^2 + 2450*x + 1291);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 44*x^10 + 63*x^9 + 574*x^8 - 851*x^7 - 2964*x^6 + 4032*x^5 + 6247*x^4 - 6815*x^3 - 4744*x^2 + 2450*x + 1291)
\( x^{12} - x^{11} - 44 x^{10} + 63 x^{9} + 574 x^{8} - 851 x^{7} - 2964 x^{6} + 4032 x^{5} + 6247 x^{4} + \cdots + 1291 \)
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: |
\(2014044676385121747377\)
\(\medspace = 17^{9}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(59.61\) | 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: | $17^{3/4}19^{2/3}\approx 59.61274106521763$ | ||
| 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$$\chi_{323}(64,·)$, $\chi_{323}(1,·)$, $\chi_{323}(273,·)$, $\chi_{323}(106,·)$, $\chi_{323}(140,·)$, $\chi_{323}(239,·)$, $\chi_{323}(305,·)$, $\chi_{323}(115,·)$, $\chi_{323}(30,·)$, $\chi_{323}(220,·)$, $\chi_{323}(254,·)$, $\chi_{323}(191,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{66\!\cdots\!14}a^{11}-\frac{408902877786881}{66\!\cdots\!14}a^{10}+\frac{832840499112269}{33\!\cdots\!07}a^{9}-\frac{815486814535982}{33\!\cdots\!07}a^{8}+\frac{32\!\cdots\!99}{66\!\cdots\!14}a^{7}+\frac{814040877792696}{33\!\cdots\!07}a^{6}-\frac{583890151524395}{66\!\cdots\!14}a^{5}+\frac{11\!\cdots\!21}{33\!\cdots\!07}a^{4}+\frac{126585318120306}{33\!\cdots\!07}a^{3}+\frac{571627441806751}{33\!\cdots\!07}a^{2}+\frac{32\!\cdots\!59}{66\!\cdots\!14}a-\frac{10\!\cdots\!91}{66\!\cdots\!14}$
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
Trivial group, which has order $1$ (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: |
$\frac{13397106688883}{33\!\cdots\!07}a^{11}-\frac{15796677132047}{33\!\cdots\!07}a^{10}-\frac{581749991818788}{33\!\cdots\!07}a^{9}+\frac{953379063047515}{33\!\cdots\!07}a^{8}+\frac{72\!\cdots\!63}{33\!\cdots\!07}a^{7}-\frac{12\!\cdots\!20}{33\!\cdots\!07}a^{6}-\frac{34\!\cdots\!41}{33\!\cdots\!07}a^{5}+\frac{60\!\cdots\!27}{33\!\cdots\!07}a^{4}+\frac{53\!\cdots\!70}{33\!\cdots\!07}a^{3}-\frac{10\!\cdots\!37}{33\!\cdots\!07}a^{2}-\frac{63\!\cdots\!37}{33\!\cdots\!07}a+\frac{27\!\cdots\!82}{33\!\cdots\!07}$, $\frac{5576464399889}{66\!\cdots\!14}a^{11}+\frac{15038979882135}{66\!\cdots\!14}a^{10}-\frac{118993668187098}{33\!\cdots\!07}a^{9}-\frac{270293117319779}{33\!\cdots\!07}a^{8}+\frac{33\!\cdots\!21}{66\!\cdots\!14}a^{7}+\frac{34\!\cdots\!33}{33\!\cdots\!07}a^{6}-\frac{19\!\cdots\!01}{66\!\cdots\!14}a^{5}-\frac{18\!\cdots\!21}{33\!\cdots\!07}a^{4}+\frac{23\!\cdots\!34}{33\!\cdots\!07}a^{3}+\frac{36\!\cdots\!31}{33\!\cdots\!07}a^{2}-\frac{46\!\cdots\!47}{66\!\cdots\!14}a-\frac{28\!\cdots\!45}{66\!\cdots\!14}$, $\frac{13397106688883}{33\!\cdots\!07}a^{11}-\frac{15796677132047}{33\!\cdots\!07}a^{10}-\frac{581749991818788}{33\!\cdots\!07}a^{9}+\frac{953379063047515}{33\!\cdots\!07}a^{8}+\frac{72\!\cdots\!63}{33\!\cdots\!07}a^{7}-\frac{12\!\cdots\!20}{33\!\cdots\!07}a^{6}-\frac{34\!\cdots\!41}{33\!\cdots\!07}a^{5}+\frac{60\!\cdots\!27}{33\!\cdots\!07}a^{4}+\frac{53\!\cdots\!70}{33\!\cdots\!07}a^{3}-\frac{10\!\cdots\!37}{33\!\cdots\!07}a^{2}-\frac{96\!\cdots\!44}{33\!\cdots\!07}a+\frac{30\!\cdots\!89}{33\!\cdots\!07}$, $\frac{16209908150820}{33\!\cdots\!07}a^{11}-\frac{10489924591142}{33\!\cdots\!07}a^{10}-\frac{694951103187666}{33\!\cdots\!07}a^{9}+\frac{771263302163741}{33\!\cdots\!07}a^{8}+\frac{86\!\cdots\!06}{33\!\cdots\!07}a^{7}-\frac{10\!\cdots\!12}{33\!\cdots\!07}a^{6}-\frac{39\!\cdots\!76}{33\!\cdots\!07}a^{5}+\frac{44\!\cdots\!16}{33\!\cdots\!07}a^{4}+\frac{57\!\cdots\!74}{33\!\cdots\!07}a^{3}-\frac{65\!\cdots\!51}{33\!\cdots\!07}a^{2}-\frac{38\!\cdots\!40}{33\!\cdots\!07}a+\frac{15\!\cdots\!12}{33\!\cdots\!07}$, $\frac{32370677777655}{66\!\cdots\!14}a^{11}-\frac{16554374381959}{66\!\cdots\!14}a^{10}-\frac{700743660005886}{33\!\cdots\!07}a^{9}+\frac{683085945727736}{33\!\cdots\!07}a^{8}+\frac{17\!\cdots\!47}{66\!\cdots\!14}a^{7}-\frac{93\!\cdots\!87}{33\!\cdots\!07}a^{6}-\frac{87\!\cdots\!83}{66\!\cdots\!14}a^{5}+\frac{42\!\cdots\!06}{33\!\cdots\!07}a^{4}+\frac{77\!\cdots\!04}{33\!\cdots\!07}a^{3}-\frac{65\!\cdots\!06}{33\!\cdots\!07}a^{2}-\frac{59\!\cdots\!21}{66\!\cdots\!14}a+\frac{32\!\cdots\!33}{66\!\cdots\!14}$, $\frac{3021501402666}{33\!\cdots\!07}a^{11}-\frac{15390252030054}{33\!\cdots\!07}a^{10}-\frac{125039972148820}{33\!\cdots\!07}a^{9}+\frac{707100547705515}{33\!\cdots\!07}a^{8}+\frac{11\!\cdots\!68}{33\!\cdots\!07}a^{7}-\frac{86\!\cdots\!44}{33\!\cdots\!07}a^{6}-\frac{12\!\cdots\!98}{33\!\cdots\!07}a^{5}+\frac{38\!\cdots\!07}{33\!\cdots\!07}a^{4}-\frac{13\!\cdots\!82}{33\!\cdots\!07}a^{3}-\frac{62\!\cdots\!76}{33\!\cdots\!07}a^{2}+\frac{29\!\cdots\!74}{33\!\cdots\!07}a+\frac{21\!\cdots\!21}{33\!\cdots\!07}$, $\frac{16001208210091}{33\!\cdots\!07}a^{11}+\frac{10207079979817}{33\!\cdots\!07}a^{10}-\frac{683112242407724}{33\!\cdots\!07}a^{9}-\frac{117953006425548}{33\!\cdots\!07}a^{8}+\frac{88\!\cdots\!81}{33\!\cdots\!07}a^{7}+\frac{12\!\cdots\!40}{33\!\cdots\!07}a^{6}-\frac{43\!\cdots\!13}{33\!\cdots\!07}a^{5}-\frac{11\!\cdots\!02}{33\!\cdots\!07}a^{4}+\frac{75\!\cdots\!60}{33\!\cdots\!07}a^{3}+\frac{33\!\cdots\!11}{33\!\cdots\!07}a^{2}-\frac{27\!\cdots\!10}{33\!\cdots\!07}a-\frac{13\!\cdots\!72}{33\!\cdots\!07}$, $\frac{675108731556}{33\!\cdots\!07}a^{11}-\frac{26820262342875}{66\!\cdots\!14}a^{10}-\frac{85365971368377}{66\!\cdots\!14}a^{9}+\frac{11\!\cdots\!97}{66\!\cdots\!14}a^{8}+\frac{12\!\cdots\!65}{66\!\cdots\!14}a^{7}-\frac{79\!\cdots\!41}{33\!\cdots\!07}a^{6}-\frac{64\!\cdots\!77}{66\!\cdots\!14}a^{5}+\frac{77\!\cdots\!49}{66\!\cdots\!14}a^{4}+\frac{67\!\cdots\!07}{66\!\cdots\!14}a^{3}-\frac{10\!\cdots\!45}{66\!\cdots\!14}a^{2}+\frac{13\!\cdots\!35}{66\!\cdots\!14}a-\frac{14\!\cdots\!33}{33\!\cdots\!07}$, $\frac{36118574727167}{33\!\cdots\!07}a^{11}+\frac{51216247746381}{66\!\cdots\!14}a^{10}-\frac{15\!\cdots\!35}{33\!\cdots\!07}a^{9}-\frac{370538826467763}{33\!\cdots\!07}a^{8}+\frac{20\!\cdots\!75}{33\!\cdots\!07}a^{7}+\frac{76\!\cdots\!95}{66\!\cdots\!14}a^{6}-\frac{20\!\cdots\!61}{66\!\cdots\!14}a^{5}-\frac{29\!\cdots\!79}{33\!\cdots\!07}a^{4}+\frac{18\!\cdots\!50}{33\!\cdots\!07}a^{3}+\frac{71\!\cdots\!37}{33\!\cdots\!07}a^{2}-\frac{72\!\cdots\!81}{33\!\cdots\!07}a-\frac{63\!\cdots\!57}{66\!\cdots\!14}$, $\frac{36215000306231}{66\!\cdots\!14}a^{11}+\frac{44519710890161}{66\!\cdots\!14}a^{10}-\frac{771356657861999}{33\!\cdots\!07}a^{9}-\frac{564202448622919}{33\!\cdots\!07}a^{8}+\frac{20\!\cdots\!93}{66\!\cdots\!14}a^{7}+\frac{61\!\cdots\!01}{33\!\cdots\!07}a^{6}-\frac{10\!\cdots\!75}{66\!\cdots\!14}a^{5}-\frac{29\!\cdots\!14}{33\!\cdots\!07}a^{4}+\frac{91\!\cdots\!15}{33\!\cdots\!07}a^{3}+\frac{44\!\cdots\!90}{33\!\cdots\!07}a^{2}-\frac{62\!\cdots\!65}{66\!\cdots\!14}a-\frac{30\!\cdots\!11}{66\!\cdots\!14}$, $\frac{42288031166631}{66\!\cdots\!14}a^{11}-\frac{35618474534441}{33\!\cdots\!07}a^{10}-\frac{18\!\cdots\!01}{66\!\cdots\!14}a^{9}+\frac{39\!\cdots\!27}{66\!\cdots\!14}a^{8}+\frac{11\!\cdots\!47}{33\!\cdots\!07}a^{7}-\frac{26\!\cdots\!44}{33\!\cdots\!07}a^{6}-\frac{48\!\cdots\!35}{33\!\cdots\!07}a^{5}+\frac{25\!\cdots\!81}{66\!\cdots\!14}a^{4}+\frac{14\!\cdots\!81}{66\!\cdots\!14}a^{3}-\frac{42\!\cdots\!81}{66\!\cdots\!14}a^{2}-\frac{10\!\cdots\!83}{33\!\cdots\!07}a+\frac{13\!\cdots\!35}{66\!\cdots\!14}$
| 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: | \( 3033543.75824 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 3033543.75824 \cdot 1}{2\cdot\sqrt{2014044676385121747377}}\cr\approx \mathstrut & 0.138434923874 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 44*x^10 + 63*x^9 + 574*x^8 - 851*x^7 - 2964*x^6 + 4032*x^5 + 6247*x^4 - 6815*x^3 - 4744*x^2 + 2450*x + 1291)
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 - 44*x^10 + 63*x^9 + 574*x^8 - 851*x^7 - 2964*x^6 + 4032*x^5 + 6247*x^4 - 6815*x^3 - 4744*x^2 + 2450*x + 1291, 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 - 44*x^10 + 63*x^9 + 574*x^8 - 851*x^7 - 2964*x^6 + 4032*x^5 + 6247*x^4 - 6815*x^3 - 4744*x^2 + 2450*x + 1291);
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 - 44*x^10 + 63*x^9 + 574*x^8 - 851*x^7 - 2964*x^6 + 4032*x^5 + 6247*x^4 - 6815*x^3 - 4744*x^2 + 2450*x + 1291);
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, 4.4.4913.1, 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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(19\)
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |