Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 + 10*x^9 - 48*x^8 + 109*x^7 - 193*x^6 + 245*x^5 - 250*x^4 + 250*x^3 - 202*x^2 + 105*x - 25)
gp: K = bnfinit(y^12 - 3*y^11 + 2*y^10 + 10*y^9 - 48*y^8 + 109*y^7 - 193*y^6 + 245*y^5 - 250*y^4 + 250*y^3 - 202*y^2 + 105*y - 25, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 2*x^10 + 10*x^9 - 48*x^8 + 109*x^7 - 193*x^6 + 245*x^5 - 250*x^4 + 250*x^3 - 202*x^2 + 105*x - 25);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 2*x^10 + 10*x^9 - 48*x^8 + 109*x^7 - 193*x^6 + 245*x^5 - 250*x^4 + 250*x^3 - 202*x^2 + 105*x - 25)
\( x^{12} - 3 x^{11} + 2 x^{10} + 10 x^{9} - 48 x^{8} + 109 x^{7} - 193 x^{6} + 245 x^{5} - 250 x^{4} + \cdots - 25 \)
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: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(860374847629609\)
\(\medspace = 13^{10}\cdot 79^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.56\) | 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: | $13^{5/6}79^{1/2}\approx 75.35284821191699$ | ||
| Ramified primes: |
\(13\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{42730785}a^{11}-\frac{2130688}{42730785}a^{10}+\frac{2219239}{14243595}a^{9}-\frac{474028}{8546157}a^{8}-\frac{2130613}{42730785}a^{7}+\frac{3535994}{42730785}a^{6}-\frac{1224312}{4747865}a^{5}+\frac{3895516}{8546157}a^{4}-\frac{168869}{949573}a^{3}+\frac{1273937}{8546157}a^{2}+\frac{3557398}{42730785}a+\frac{4055626}{8546157}$
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 $1$
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: | $7$ | 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: |
$\frac{9827812}{42730785}a^{11}-\frac{22046521}{42730785}a^{10}+\frac{972433}{14243595}a^{9}+\frac{20188385}{8546157}a^{8}-\frac{395961031}{42730785}a^{7}+\frac{771889703}{42730785}a^{6}-\frac{145237854}{4747865}a^{5}+\frac{279083257}{8546157}a^{4}-\frac{91325945}{2848719}a^{3}+\frac{275059190}{8546157}a^{2}-\frac{918021014}{42730785}a+\frac{63738217}{8546157}$, $\frac{4931947}{42730785}a^{11}-\frac{14424361}{42730785}a^{10}+\frac{1405403}{14243595}a^{9}+\frac{11433842}{8546157}a^{8}-\frac{228515731}{42730785}a^{7}+\frac{474333968}{42730785}a^{6}-\frac{253326272}{14243595}a^{5}+\frac{179105761}{8546157}a^{4}-\frac{49043240}{2848719}a^{3}+\frac{165649310}{8546157}a^{2}-\frac{589824614}{42730785}a+\frac{20099743}{8546157}$, $\frac{2853097}{14243595}a^{11}-\frac{7143496}{14243595}a^{10}+\frac{1291364}{14243595}a^{9}+\frac{6169432}{2848719}a^{8}-\frac{120763906}{14243595}a^{7}+\frac{241834228}{14243595}a^{6}-\frac{399904111}{14243595}a^{5}+\frac{29832610}{949573}a^{4}-\frac{27260535}{949573}a^{3}+\frac{28638509}{949573}a^{2}-\frac{98008653}{4747865}a+\frac{13708696}{2848719}$, $\frac{4300589}{42730785}a^{11}-\frac{9596237}{42730785}a^{10}+\frac{53261}{14243595}a^{9}+\frac{9234445}{8546157}a^{8}-\frac{171334292}{42730785}a^{7}+\frac{323660491}{42730785}a^{6}-\frac{177813049}{14243595}a^{5}+\frac{109884179}{8546157}a^{4}-\frac{11279598}{949573}a^{3}+\frac{102832225}{8546157}a^{2}-\frac{352336003}{42730785}a+\frac{13614314}{8546157}$, $\frac{11551549}{42730785}a^{11}-\frac{28329637}{42730785}a^{10}+\frac{676962}{4747865}a^{9}+\frac{24353957}{8546157}a^{8}-\frac{486617797}{42730785}a^{7}+\frac{975817211}{42730785}a^{6}-\frac{182357273}{4747865}a^{5}+\frac{368709583}{8546157}a^{4}-\frac{115528066}{2848719}a^{3}+\frac{360670898}{8546157}a^{2}-\frac{1222907228}{42730785}a+\frac{80083480}{8546157}$, $\frac{3565408}{42730785}a^{11}-\frac{7621834}{42730785}a^{10}-\frac{208993}{14243595}a^{9}+\frac{7574648}{8546157}a^{8}-\frac{139036894}{42730785}a^{7}+\frac{255361052}{42730785}a^{6}-\frac{46440541}{4747865}a^{5}+\frac{83267020}{8546157}a^{4}-\frac{25620841}{2848719}a^{3}+\frac{84807944}{8546157}a^{2}-\frac{246353951}{42730785}a+\frac{11874796}{8546157}$, $\frac{25628}{2848719}a^{11}-\frac{76699}{2848719}a^{10}-\frac{17743}{949573}a^{9}+\frac{337744}{2848719}a^{8}-\frac{1003318}{2848719}a^{7}+\frac{584423}{949573}a^{6}-\frac{731464}{949573}a^{5}+\frac{1227964}{949573}a^{4}-\frac{1936370}{2848719}a^{3}+\frac{6890188}{2848719}a^{2}-\frac{1102026}{949573}a-\frac{1042811}{2848719}$
| 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: | \( 524.301439397 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 524.301439397 \cdot 1}{2\cdot\sqrt{860374847629609}}\cr\approx \mathstrut & 0.222867456987 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 + 10*x^9 - 48*x^8 + 109*x^7 - 193*x^6 + 245*x^5 - 250*x^4 + 250*x^3 - 202*x^2 + 105*x - 25)
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 - 3*x^11 + 2*x^10 + 10*x^9 - 48*x^8 + 109*x^7 - 193*x^6 + 245*x^5 - 250*x^4 + 250*x^3 - 202*x^2 + 105*x - 25, 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 - 3*x^11 + 2*x^10 + 10*x^9 - 48*x^8 + 109*x^7 - 193*x^6 + 245*x^5 - 250*x^4 + 250*x^3 - 202*x^2 + 105*x - 25);
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 - 3*x^11 + 2*x^10 + 10*x^9 - 48*x^8 + 109*x^7 - 193*x^6 + 245*x^5 - 250*x^4 + 250*x^3 - 202*x^2 + 105*x - 25);
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
$C_2^3:A_4$ (as 12T58):
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 96 |
| The 10 conjugacy class representatives for $C_2^3:A_4$ |
| Character table for $C_2^3:A_4$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{13})^+\) |
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 8 siblings: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Minimal sibling: | 8.0.30124114969.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
|
\(79\)
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 79.2.1.1 | $x^{2} + 237$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |