Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 17*x^10 + 68*x^9 + 108*x^8 - 416*x^7 - 314*x^6 + 1129*x^5 + 358*x^4 - 1353*x^3 - 36*x^2 + 540*x - 72)
gp: K = bnfinit(y^12 - 4*y^11 - 17*y^10 + 68*y^9 + 108*y^8 - 416*y^7 - 314*y^6 + 1129*y^5 + 358*y^4 - 1353*y^3 - 36*y^2 + 540*y - 72, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 17*x^10 + 68*x^9 + 108*x^8 - 416*x^7 - 314*x^6 + 1129*x^5 + 358*x^4 - 1353*x^3 - 36*x^2 + 540*x - 72);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 17*x^10 + 68*x^9 + 108*x^8 - 416*x^7 - 314*x^6 + 1129*x^5 + 358*x^4 - 1353*x^3 - 36*x^2 + 540*x - 72)
\( x^{12} - 4 x^{11} - 17 x^{10} + 68 x^{9} + 108 x^{8} - 416 x^{7} - 314 x^{6} + 1129 x^{5} + 358 x^{4} + \cdots - 72 \)
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: |
\(139754631175017849\)
\(\medspace = 3^{6}\cdot 61^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.84\) | 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: | $3^{1/2}61^{2/3}\approx 26.839876760292654$ | ||
| Ramified primes: |
\(3\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{28}a^{9}+\frac{1}{7}a^{8}-\frac{5}{14}a^{7}+\frac{3}{14}a^{6}+\frac{5}{14}a^{5}-\frac{1}{2}a^{4}+\frac{1}{7}a^{3}+\frac{3}{28}a^{2}+\frac{1}{14}a+\frac{3}{7}$, $\frac{1}{168}a^{10}+\frac{1}{84}a^{9}+\frac{5}{84}a^{8}-\frac{29}{84}a^{7}+\frac{9}{28}a^{6}+\frac{11}{84}a^{5}+\frac{1}{42}a^{4}-\frac{5}{168}a^{3}+\frac{13}{42}a^{2}+\frac{3}{14}a-\frac{1}{7}$, $\frac{1}{71232}a^{11}+\frac{37}{23744}a^{10}+\frac{73}{5936}a^{9}-\frac{605}{2968}a^{8}+\frac{4457}{17808}a^{7}+\frac{5347}{17808}a^{6}-\frac{127}{672}a^{5}-\frac{17209}{71232}a^{4}-\frac{31669}{71232}a^{3}-\frac{4063}{8904}a^{2}-\frac{1}{5936}a-\frac{247}{2968}$
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
Trivial group, which has order $1$
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: |
$\frac{113}{4452}a^{11}-\frac{109}{742}a^{10}-\frac{235}{1484}a^{9}+\frac{1451}{742}a^{8}-\frac{1025}{1113}a^{7}-\frac{8893}{1113}a^{6}+\frac{313}{42}a^{5}+\frac{43837}{4452}a^{4}-\frac{14306}{1113}a^{3}+\frac{6467}{4452}a^{2}+\frac{2265}{742}a-\frac{702}{371}$, $\frac{845}{35616}a^{11}-\frac{6481}{35616}a^{10}+\frac{403}{8904}a^{9}+\frac{5399}{2226}a^{8}-\frac{4651}{1272}a^{7}-\frac{91195}{8904}a^{6}+\frac{2307}{112}a^{5}+\frac{177481}{11872}a^{4}-\frac{459535}{11872}a^{3}-\frac{301}{159}a^{2}+\frac{8753}{424}a-\frac{6467}{1484}$, $\frac{299}{23744}a^{11}-\frac{3041}{71232}a^{10}-\frac{4321}{17808}a^{9}+\frac{863}{1272}a^{8}+\frac{35225}{17808}a^{7}-\frac{22623}{5936}a^{6}-\frac{5471}{672}a^{5}+\frac{643303}{71232}a^{4}+\frac{1056667}{71232}a^{3}-\frac{74749}{8904}a^{2}-\frac{49233}{5936}a+\frac{6553}{2968}$, $\frac{299}{23744}a^{11}-\frac{3041}{71232}a^{10}-\frac{4321}{17808}a^{9}+\frac{863}{1272}a^{8}+\frac{35225}{17808}a^{7}-\frac{22623}{5936}a^{6}-\frac{5471}{672}a^{5}+\frac{643303}{71232}a^{4}+\frac{1056667}{71232}a^{3}-\frac{74749}{8904}a^{2}-\frac{49233}{5936}a+\frac{3585}{2968}$, $\frac{845}{35616}a^{11}-\frac{6481}{35616}a^{10}+\frac{403}{8904}a^{9}+\frac{5399}{2226}a^{8}-\frac{4651}{1272}a^{7}-\frac{91195}{8904}a^{6}+\frac{2307}{112}a^{5}+\frac{177481}{11872}a^{4}-\frac{459535}{11872}a^{3}-\frac{301}{159}a^{2}+\frac{8753}{424}a-\frac{4983}{1484}$, $\frac{5015}{71232}a^{11}-\frac{7365}{23744}a^{10}-\frac{5541}{5936}a^{9}+\frac{1933}{424}a^{8}+\frac{70867}{17808}a^{7}-\frac{393511}{17808}a^{6}-\frac{4097}{672}a^{5}+\frac{2843713}{71232}a^{4}+\frac{41317}{71232}a^{3}-\frac{177857}{8904}a^{2}+\frac{8553}{5936}a-\frac{1119}{424}$, $\frac{1213}{71232}a^{11}-\frac{4429}{71232}a^{10}-\frac{4229}{17808}a^{9}+\frac{6385}{8904}a^{8}+\frac{7887}{5936}a^{7}-\frac{30545}{17808}a^{6}-\frac{3155}{672}a^{5}-\frac{239285}{71232}a^{4}+\frac{746143}{71232}a^{3}+\frac{27365}{2968}a^{2}-\frac{68205}{5936}a+\frac{2701}{2968}$, $\frac{341}{10176}a^{11}-\frac{1569}{23744}a^{10}-\frac{4465}{5936}a^{9}+\frac{395}{424}a^{8}+\frac{113059}{17808}a^{7}-\frac{69919}{17808}a^{6}-\frac{15965}{672}a^{5}+\frac{333349}{71232}a^{4}+\frac{2532337}{71232}a^{3}-\frac{14963}{8904}a^{2}-\frac{102451}{5936}a+\frac{2315}{2968}$, $\frac{727}{71232}a^{11}-\frac{1313}{10176}a^{10}+\frac{4877}{17808}a^{9}+\frac{2057}{1272}a^{8}-\frac{96641}{17808}a^{7}-\frac{101723}{17808}a^{6}+\frac{6365}{224}a^{5}+\frac{78731}{23744}a^{4}-\frac{184791}{3392}a^{3}+\frac{82039}{8904}a^{2}+\frac{184985}{5936}a-\frac{11665}{2968}$, $\frac{89}{5088}a^{11}-\frac{1541}{11872}a^{10}-\frac{101}{2968}a^{9}+\frac{2883}{1484}a^{8}-\frac{16601}{8904}a^{7}-\frac{89227}{8904}a^{6}+\frac{3967}{336}a^{5}+\frac{772585}{35616}a^{4}-\frac{114269}{5088}a^{3}-\frac{84239}{4452}a^{2}+\frac{4787}{424}a+\frac{9571}{1484}$, $\frac{6563}{71232}a^{11}-\frac{9449}{23744}a^{10}-\frac{7653}{5936}a^{9}+\frac{17635}{2968}a^{8}+\frac{112951}{17808}a^{7}-\frac{528091}{17808}a^{6}-\frac{9797}{672}a^{5}+\frac{4091509}{71232}a^{4}+\frac{1159945}{71232}a^{3}-\frac{343019}{8904}a^{2}-\frac{59987}{5936}a+\frac{14307}{2968}$
| 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: | \( 55324.635116 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 55324.635116 \cdot 1}{2\cdot\sqrt{139754631175017849}}\cr\approx \mathstrut & 0.30308567219 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 17*x^10 + 68*x^9 + 108*x^8 - 416*x^7 - 314*x^6 + 1129*x^5 + 358*x^4 - 1353*x^3 - 36*x^2 + 540*x - 72)
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 - 4*x^11 - 17*x^10 + 68*x^9 + 108*x^8 - 416*x^7 - 314*x^6 + 1129*x^5 + 358*x^4 - 1353*x^3 - 36*x^2 + 540*x - 72, 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 - 4*x^11 - 17*x^10 + 68*x^9 + 108*x^8 - 416*x^7 - 314*x^6 + 1129*x^5 + 358*x^4 - 1353*x^3 - 36*x^2 + 540*x - 72);
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 - 4*x^11 - 17*x^10 + 68*x^9 + 108*x^8 - 416*x^7 - 314*x^6 + 1129*x^5 + 358*x^4 - 1353*x^3 - 36*x^2 + 540*x - 72);
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 solvable group of order 12 |
| The 4 conjugacy class representatives for $A_4$ |
| Character table for $A_4$ |
Intermediate fields
| 3.3.3721.1, 4.4.33489.1 x4, 6.6.124612569.1 x3 |
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]
Sibling fields
| Degree 4 sibling: | 4.4.33489.1 |
| Degree 6 sibling: | 6.6.124612569.1 |
| Minimal sibling: | 4.4.33489.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(61\)
| 61.3.2.1 | $x^{3} + 61$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 61.3.2.1 | $x^{3} + 61$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 61.3.2.1 | $x^{3} + 61$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 61.3.2.1 | $x^{3} + 61$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |