Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 16*x^10 + 90*x^8 - 230*x^6 + 273*x^4 - 126*x^2 + 7)
gp: K = bnfinit(y^12 - 16*y^10 + 90*y^8 - 230*y^6 + 273*y^4 - 126*y^2 + 7, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 16*x^10 + 90*x^8 - 230*x^6 + 273*x^4 - 126*x^2 + 7);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 16*x^10 + 90*x^8 - 230*x^6 + 273*x^4 - 126*x^2 + 7)
\( x^{12} - 16x^{10} + 90x^{8} - 230x^{6} + 273x^{4} - 126x^{2} + 7 \)
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: |
\(43329355585159168\)
\(\medspace = 2^{30}\cdot 7^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.34\) | 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: | $2^{11/4}7^{5/6}\approx 34.04715710793443$ | ||
| Ramified primes: |
\(2\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{113}a^{10}+\frac{54}{113}a^{8}+\frac{28}{113}a^{6}+\frac{35}{113}a^{4}+\frac{11}{113}a^{2}-\frac{34}{113}$, $\frac{1}{113}a^{11}+\frac{54}{113}a^{9}+\frac{28}{113}a^{7}+\frac{35}{113}a^{5}+\frac{11}{113}a^{3}-\frac{34}{113}a$
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: |
$\frac{2}{113}a^{10}-\frac{5}{113}a^{8}-\frac{170}{113}a^{6}+\frac{861}{113}a^{4}-\frac{1108}{113}a^{2}+\frac{158}{113}$, $\frac{48}{113}a^{10}-\frac{685}{113}a^{8}+\frac{3152}{113}a^{6}-\frac{5778}{113}a^{4}+\frac{3692}{113}a^{2}-\frac{163}{113}$, $\frac{10}{113}a^{10}-\frac{138}{113}a^{8}+\frac{619}{113}a^{6}-\frac{1232}{113}a^{4}+\frac{1127}{113}a^{2}-\frac{114}{113}$, $\frac{8}{113}a^{10}-\frac{133}{113}a^{8}+\frac{789}{113}a^{6}-\frac{2093}{113}a^{4}+\frac{2235}{113}a^{2}-\frac{498}{113}$, $\frac{75}{113}a^{10}-\frac{1035}{113}a^{8}+\frac{4473}{113}a^{6}-\frac{7432}{113}a^{4}+\frac{4328}{113}a^{2}-\frac{290}{113}$, $\frac{41}{113}a^{11}+\frac{13}{113}a^{10}-\frac{611}{113}a^{9}-\frac{202}{113}a^{8}+\frac{3069}{113}a^{7}+\frac{1042}{113}a^{6}-\frac{6701}{113}a^{5}-\frac{2144}{113}a^{4}+\frac{6327}{113}a^{3}+\frac{1725}{113}a^{2}-\frac{1959}{113}a-\frac{442}{113}$, $\frac{33}{113}a^{11}+\frac{40}{113}a^{10}-\frac{478}{113}a^{9}-\frac{552}{113}a^{8}+\frac{2280}{113}a^{7}+\frac{2363}{113}a^{6}-\frac{4608}{113}a^{5}-\frac{3685}{113}a^{4}+\frac{4092}{113}a^{3}+\frac{1457}{113}a^{2}-\frac{1461}{113}a+\frac{335}{113}$, $\frac{71}{113}a^{11}-\frac{2}{113}a^{10}-\frac{1025}{113}a^{9}+\frac{5}{113}a^{8}+\frac{4813}{113}a^{7}+\frac{170}{113}a^{6}-\frac{9154}{113}a^{5}-\frac{861}{113}a^{4}+\frac{6544}{113}a^{3}+\frac{1108}{113}a^{2}-\frac{1058}{113}a-\frac{271}{113}$, $\frac{38}{113}a^{11}-\frac{2}{113}a^{10}-\frac{547}{113}a^{9}+\frac{5}{113}a^{8}+\frac{2533}{113}a^{7}+\frac{170}{113}a^{6}-\frac{4546}{113}a^{5}-\frac{861}{113}a^{4}+\frac{2452}{113}a^{3}+\frac{1108}{113}a^{2}+\frac{403}{113}a-\frac{158}{113}$, $\frac{50}{113}a^{11}+\frac{48}{113}a^{10}-\frac{690}{113}a^{9}-\frac{685}{113}a^{8}+\frac{2982}{113}a^{7}+\frac{3152}{113}a^{6}-\frac{4917}{113}a^{5}-\frac{5778}{113}a^{4}+\frac{2584}{113}a^{3}+\frac{3692}{113}a^{2}-\frac{5}{113}a-\frac{276}{113}$, $\frac{42}{113}a^{11}+\frac{10}{113}a^{10}-\frac{557}{113}a^{9}-\frac{138}{113}a^{8}+\frac{2193}{113}a^{7}+\frac{619}{113}a^{6}-\frac{2824}{113}a^{5}-\frac{1232}{113}a^{4}+\frac{349}{113}a^{3}+\frac{1127}{113}a^{2}+\frac{493}{113}a-\frac{114}{113}$
| 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: | \( 21217.6374054 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 21217.6374054 \cdot 1}{2\cdot\sqrt{43329355585159168}}\cr\approx \mathstrut & 0.208754506988 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 16*x^10 + 90*x^8 - 230*x^6 + 273*x^4 - 126*x^2 + 7)
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 - 16*x^10 + 90*x^8 - 230*x^6 + 273*x^4 - 126*x^2 + 7, 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 - 16*x^10 + 90*x^8 - 230*x^6 + 273*x^4 - 126*x^2 + 7);
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 - 16*x^10 + 90*x^8 - 230*x^6 + 273*x^4 - 126*x^2 + 7);
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
$C_3\times D_4$ (as 12T14):
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 24 |
| The 15 conjugacy class representatives for $D_4 \times C_3$ |
| Character table for $D_4 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 4.4.7168.1, 6.6.1229312.1 |
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]
Sibling fields
| Galois closure: | deg 24 |
| Degree 12 sibling: | 12.12.37913186137014272.1 |
| Minimal sibling: | 12.12.37913186137014272.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.30.256 | $x^{12} + 12 x^{11} + 88 x^{10} + 396 x^{9} + 1308 x^{8} + 3036 x^{7} + 6126 x^{6} + 8928 x^{5} + 9281 x^{4} - 1260 x^{3} - 2242 x^{2} - 6960 x + 3767$ | $4$ | $3$ | $30$ | $D_4 \times C_3$ | $[2, 3, 7/2]^{3}$ |
|
\(7\)
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |