Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 68*x^10 + 3*x^9 + 1528*x^8 + 1135*x^7 - 12600*x^6 - 17100*x^5 + 25453*x^4 + 41563*x^3 - 10018*x^2 - 22360*x - 2789)
gp: K = bnfinit(y^12 - y^11 - 68*y^10 + 3*y^9 + 1528*y^8 + 1135*y^7 - 12600*y^6 - 17100*y^5 + 25453*y^4 + 41563*y^3 - 10018*y^2 - 22360*y - 2789, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 68*x^10 + 3*x^9 + 1528*x^8 + 1135*x^7 - 12600*x^6 - 17100*x^5 + 25453*x^4 + 41563*x^3 - 10018*x^2 - 22360*x - 2789);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 68*x^10 + 3*x^9 + 1528*x^8 + 1135*x^7 - 12600*x^6 - 17100*x^5 + 25453*x^4 + 41563*x^3 - 10018*x^2 - 22360*x - 2789)
\( x^{12} - x^{11} - 68 x^{10} + 3 x^{9} + 1528 x^{8} + 1135 x^{7} - 12600 x^{6} - 17100 x^{5} + \cdots - 2789 \)
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: |
\(416537479679833550394737\)
\(\medspace = 17^{9}\cdot 37^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(92.96\) | 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: | $17^{3/4}37^{2/3}\approx 92.96179631764876$ | ||
| Ramified primes: |
\(17\), \(37\)
| 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: | \(629=17\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{629}(1,·)$, $\chi_{629}(322,·)$, $\chi_{629}(38,·)$, $\chi_{629}(137,·)$, $\chi_{629}(174,·)$, $\chi_{629}(47,·)$, $\chi_{629}(528,·)$, $\chi_{629}(84,·)$, $\chi_{629}(149,·)$, $\chi_{629}(186,·)$, $\chi_{629}(285,·)$, $\chi_{629}(565,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{53\!\cdots\!02}a^{11}+\frac{93\!\cdots\!57}{53\!\cdots\!02}a^{10}+\frac{58\!\cdots\!84}{26\!\cdots\!51}a^{9}+\frac{99\!\cdots\!14}{26\!\cdots\!51}a^{8}-\frac{12\!\cdots\!91}{53\!\cdots\!02}a^{7}+\frac{45\!\cdots\!72}{26\!\cdots\!51}a^{6}-\frac{25\!\cdots\!61}{53\!\cdots\!02}a^{5}+\frac{19\!\cdots\!14}{26\!\cdots\!51}a^{4}+\frac{13\!\cdots\!57}{26\!\cdots\!51}a^{3}-\frac{60\!\cdots\!85}{26\!\cdots\!51}a^{2}-\frac{68\!\cdots\!93}{53\!\cdots\!02}a+\frac{21\!\cdots\!79}{53\!\cdots\!02}$
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: |
$\frac{42\!\cdots\!47}{26\!\cdots\!51}a^{11}-\frac{10\!\cdots\!21}{26\!\cdots\!51}a^{10}-\frac{26\!\cdots\!28}{26\!\cdots\!51}a^{9}+\frac{38\!\cdots\!11}{26\!\cdots\!51}a^{8}+\frac{53\!\cdots\!57}{26\!\cdots\!51}a^{7}-\frac{30\!\cdots\!46}{26\!\cdots\!51}a^{6}-\frac{37\!\cdots\!77}{26\!\cdots\!51}a^{5}-\frac{56\!\cdots\!91}{26\!\cdots\!51}a^{4}+\frac{38\!\cdots\!08}{26\!\cdots\!51}a^{3}-\frac{31\!\cdots\!43}{26\!\cdots\!51}a^{2}+\frac{20\!\cdots\!09}{26\!\cdots\!51}a+\frac{58\!\cdots\!10}{26\!\cdots\!51}$, $\frac{30\!\cdots\!65}{53\!\cdots\!02}a^{11}-\frac{71\!\cdots\!83}{53\!\cdots\!02}a^{10}-\frac{10\!\cdots\!44}{26\!\cdots\!51}a^{9}+\frac{13\!\cdots\!24}{26\!\cdots\!51}a^{8}+\frac{43\!\cdots\!87}{53\!\cdots\!02}a^{7}-\frac{10\!\cdots\!44}{26\!\cdots\!51}a^{6}-\frac{35\!\cdots\!21}{53\!\cdots\!02}a^{5}-\frac{44\!\cdots\!42}{26\!\cdots\!51}a^{4}+\frac{44\!\cdots\!35}{26\!\cdots\!51}a^{3}+\frac{18\!\cdots\!78}{26\!\cdots\!51}a^{2}-\frac{63\!\cdots\!39}{53\!\cdots\!02}a-\frac{24\!\cdots\!71}{53\!\cdots\!02}$, $\frac{35\!\cdots\!47}{53\!\cdots\!02}a^{11}-\frac{97\!\cdots\!33}{53\!\cdots\!02}a^{10}-\frac{11\!\cdots\!42}{26\!\cdots\!51}a^{9}+\frac{20\!\cdots\!38}{26\!\cdots\!51}a^{8}+\frac{45\!\cdots\!69}{53\!\cdots\!02}a^{7}-\frac{21\!\cdots\!24}{26\!\cdots\!51}a^{6}-\frac{35\!\cdots\!79}{53\!\cdots\!02}a^{5}+\frac{30\!\cdots\!41}{26\!\cdots\!51}a^{4}+\frac{36\!\cdots\!81}{26\!\cdots\!51}a^{3}-\frac{48\!\cdots\!49}{26\!\cdots\!51}a^{2}-\frac{27\!\cdots\!19}{53\!\cdots\!02}a+\frac{94\!\cdots\!27}{53\!\cdots\!02}$, $\frac{42\!\cdots\!47}{26\!\cdots\!51}a^{11}-\frac{10\!\cdots\!21}{26\!\cdots\!51}a^{10}-\frac{26\!\cdots\!28}{26\!\cdots\!51}a^{9}+\frac{38\!\cdots\!11}{26\!\cdots\!51}a^{8}+\frac{53\!\cdots\!57}{26\!\cdots\!51}a^{7}-\frac{30\!\cdots\!46}{26\!\cdots\!51}a^{6}-\frac{37\!\cdots\!77}{26\!\cdots\!51}a^{5}-\frac{56\!\cdots\!91}{26\!\cdots\!51}a^{4}+\frac{38\!\cdots\!08}{26\!\cdots\!51}a^{3}-\frac{31\!\cdots\!43}{26\!\cdots\!51}a^{2}-\frac{61\!\cdots\!42}{26\!\cdots\!51}a+\frac{48\!\cdots\!08}{26\!\cdots\!51}$, $\frac{14\!\cdots\!39}{26\!\cdots\!51}a^{11}-\frac{58\!\cdots\!11}{26\!\cdots\!51}a^{10}-\frac{84\!\cdots\!38}{26\!\cdots\!51}a^{9}+\frac{28\!\cdots\!32}{26\!\cdots\!51}a^{8}+\frac{16\!\cdots\!23}{26\!\cdots\!51}a^{7}-\frac{42\!\cdots\!26}{26\!\cdots\!51}a^{6}-\frac{13\!\cdots\!97}{26\!\cdots\!51}a^{5}+\frac{21\!\cdots\!49}{26\!\cdots\!51}a^{4}+\frac{33\!\cdots\!54}{26\!\cdots\!51}a^{3}-\frac{32\!\cdots\!86}{26\!\cdots\!51}a^{2}-\frac{17\!\cdots\!02}{26\!\cdots\!51}a-\frac{12\!\cdots\!23}{26\!\cdots\!51}$, $\frac{46\!\cdots\!39}{26\!\cdots\!51}a^{11}-\frac{42\!\cdots\!87}{53\!\cdots\!02}a^{10}-\frac{33\!\cdots\!88}{26\!\cdots\!51}a^{9}-\frac{12\!\cdots\!77}{26\!\cdots\!51}a^{8}+\frac{77\!\cdots\!54}{26\!\cdots\!51}a^{7}+\frac{15\!\cdots\!83}{53\!\cdots\!02}a^{6}-\frac{13\!\cdots\!39}{53\!\cdots\!02}a^{5}-\frac{92\!\cdots\!61}{26\!\cdots\!51}a^{4}+\frac{15\!\cdots\!54}{26\!\cdots\!51}a^{3}+\frac{21\!\cdots\!39}{26\!\cdots\!51}a^{2}-\frac{96\!\cdots\!71}{26\!\cdots\!51}a-\frac{22\!\cdots\!77}{53\!\cdots\!02}$, $\frac{15\!\cdots\!13}{53\!\cdots\!02}a^{11}-\frac{15\!\cdots\!51}{26\!\cdots\!51}a^{10}-\frac{51\!\cdots\!04}{26\!\cdots\!51}a^{9}+\frac{56\!\cdots\!77}{26\!\cdots\!51}a^{8}+\frac{22\!\cdots\!91}{53\!\cdots\!02}a^{7}-\frac{66\!\cdots\!71}{53\!\cdots\!02}a^{6}-\frac{96\!\cdots\!75}{26\!\cdots\!51}a^{5}-\frac{29\!\cdots\!63}{26\!\cdots\!51}a^{4}+\frac{24\!\cdots\!87}{26\!\cdots\!51}a^{3}+\frac{64\!\cdots\!73}{26\!\cdots\!51}a^{2}-\frac{40\!\cdots\!55}{53\!\cdots\!02}a+\frac{20\!\cdots\!15}{26\!\cdots\!51}$, $\frac{10\!\cdots\!73}{53\!\cdots\!02}a^{11}+\frac{20\!\cdots\!77}{53\!\cdots\!02}a^{10}-\frac{73\!\cdots\!79}{53\!\cdots\!02}a^{9}-\frac{66\!\cdots\!71}{53\!\cdots\!02}a^{8}+\frac{83\!\cdots\!76}{26\!\cdots\!51}a^{7}+\frac{26\!\cdots\!33}{53\!\cdots\!02}a^{6}-\frac{13\!\cdots\!45}{53\!\cdots\!02}a^{5}-\frac{28\!\cdots\!35}{53\!\cdots\!02}a^{4}+\frac{15\!\cdots\!63}{53\!\cdots\!02}a^{3}+\frac{52\!\cdots\!17}{53\!\cdots\!02}a^{2}+\frac{76\!\cdots\!92}{26\!\cdots\!51}a+\frac{59\!\cdots\!45}{26\!\cdots\!51}$, $\frac{29\!\cdots\!87}{53\!\cdots\!02}a^{11}+\frac{16\!\cdots\!67}{26\!\cdots\!51}a^{10}-\frac{30\!\cdots\!31}{53\!\cdots\!02}a^{9}-\frac{22\!\cdots\!65}{53\!\cdots\!02}a^{8}+\frac{45\!\cdots\!92}{26\!\cdots\!51}a^{7}+\frac{25\!\cdots\!08}{26\!\cdots\!51}a^{6}-\frac{43\!\cdots\!34}{26\!\cdots\!51}a^{5}-\frac{41\!\cdots\!21}{53\!\cdots\!02}a^{4}+\frac{18\!\cdots\!49}{53\!\cdots\!02}a^{3}+\frac{92\!\cdots\!05}{53\!\cdots\!02}a^{2}-\frac{60\!\cdots\!16}{26\!\cdots\!51}a-\frac{47\!\cdots\!73}{53\!\cdots\!02}$, $\frac{43\!\cdots\!33}{53\!\cdots\!02}a^{11}-\frac{52\!\cdots\!60}{26\!\cdots\!51}a^{10}-\frac{13\!\cdots\!92}{26\!\cdots\!51}a^{9}+\frac{20\!\cdots\!06}{26\!\cdots\!51}a^{8}+\frac{59\!\cdots\!87}{53\!\cdots\!02}a^{7}-\frac{34\!\cdots\!35}{53\!\cdots\!02}a^{6}-\frac{24\!\cdots\!29}{26\!\cdots\!51}a^{5}-\frac{34\!\cdots\!15}{26\!\cdots\!51}a^{4}+\frac{54\!\cdots\!48}{26\!\cdots\!51}a^{3}+\frac{14\!\cdots\!73}{26\!\cdots\!51}a^{2}-\frac{59\!\cdots\!51}{53\!\cdots\!02}a-\frac{80\!\cdots\!95}{26\!\cdots\!51}$, $\frac{11\!\cdots\!33}{53\!\cdots\!02}a^{11}+\frac{10\!\cdots\!15}{53\!\cdots\!02}a^{10}-\frac{10\!\cdots\!85}{53\!\cdots\!02}a^{9}-\frac{71\!\cdots\!31}{53\!\cdots\!02}a^{8}+\frac{15\!\cdots\!12}{26\!\cdots\!51}a^{7}+\frac{16\!\cdots\!85}{53\!\cdots\!02}a^{6}-\frac{29\!\cdots\!59}{53\!\cdots\!02}a^{5}-\frac{13\!\cdots\!25}{53\!\cdots\!02}a^{4}+\frac{64\!\cdots\!55}{53\!\cdots\!02}a^{3}+\frac{32\!\cdots\!63}{53\!\cdots\!02}a^{2}-\frac{25\!\cdots\!32}{26\!\cdots\!51}a-\frac{86\!\cdots\!08}{26\!\cdots\!51}$
| 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: | \( 65606044.7889 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 65606044.7889 \cdot 1}{2\cdot\sqrt{416537479679833550394737}}\cr\approx \mathstrut & 0.208183720640 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 68*x^10 + 3*x^9 + 1528*x^8 + 1135*x^7 - 12600*x^6 - 17100*x^5 + 25453*x^4 + 41563*x^3 - 10018*x^2 - 22360*x - 2789)
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 - 68*x^10 + 3*x^9 + 1528*x^8 + 1135*x^7 - 12600*x^6 - 17100*x^5 + 25453*x^4 + 41563*x^3 - 10018*x^2 - 22360*x - 2789, 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 - 68*x^10 + 3*x^9 + 1528*x^8 + 1135*x^7 - 12600*x^6 - 17100*x^5 + 25453*x^4 + 41563*x^3 - 10018*x^2 - 22360*x - 2789);
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 - 68*x^10 + 3*x^9 + 1528*x^8 + 1135*x^7 - 12600*x^6 - 17100*x^5 + 25453*x^4 + 41563*x^3 - 10018*x^2 - 22360*x - 2789);
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
| \(\Q(\sqrt{17}) \), 3.3.1369.1, 4.4.4913.1, 6.6.9207752993.1 |
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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.1.0.1}{1} }^{12}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(37\)
| 37.12.8.1 | $x^{12} + 18 x^{10} + 220 x^{9} + 114 x^{8} + 864 x^{7} - 5754 x^{6} + 7320 x^{5} - 47346 x^{4} - 240044 x^{3} + 340080 x^{2} - 2045220 x + 8612757$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |