Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 3*x^6 - 36*x^5 + 143*x^4 + 204*x^3 - 1476*x^2 + 1800*x - 432)
gp: K = bnfinit(y^7 - 3*y^6 - 36*y^5 + 143*y^4 + 204*y^3 - 1476*y^2 + 1800*y - 432, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^7 - 3*x^6 - 36*x^5 + 143*x^4 + 204*x^3 - 1476*x^2 + 1800*x - 432);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^7 - 3*x^6 - 36*x^5 + 143*x^4 + 204*x^3 - 1476*x^2 + 1800*x - 432)
\( x^{7} - 3x^{6} - 36x^{5} + 143x^{4} + 204x^{3} - 1476x^{2} + 1800x - 432 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[7, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(58280536320561\)
\(\medspace = 3^{8}\cdot 307^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(92.58\) | 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: | $3^{4/3}307^{2/3}\approx 196.903720979923$ | ||
| Ramified primes: |
\(3\), \(307\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{12}a^{5}-\frac{1}{4}a^{4}-\frac{1}{12}a^{2}$, $\frac{1}{72}a^{6}-\frac{1}{24}a^{5}+\frac{35}{72}a^{3}-\frac{1}{6}a^{2}$
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: | $6$ | 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: |
$\frac{1}{6}a^{6}-\frac{11}{2}a^{4}+\frac{41}{6}a^{3}+\frac{83}{2}a^{2}-95a+49$, $\frac{1}{72}a^{6}+\frac{1}{8}a^{5}-a^{4}-\frac{145}{72}a^{3}+\frac{56}{3}a^{2}-\frac{51}{2}a+5$, $\frac{23}{72}a^{6}-\frac{31}{24}a^{5}-\frac{13}{2}a^{4}+\frac{2713}{72}a^{3}-\frac{95}{2}a^{2}+\frac{17}{2}a+1$, $\frac{7}{24}a^{6}+\frac{5}{24}a^{5}-\frac{39}{4}a^{4}+\frac{137}{24}a^{3}+\frac{965}{12}a^{2}-\frac{265}{2}a+34$, $\frac{1}{3}a^{6}-\frac{5}{12}a^{5}-\frac{47}{4}a^{4}+\frac{71}{3}a^{3}+\frac{1073}{12}a^{2}-232a+64$, $\frac{7}{6}a^{6}+\frac{1}{3}a^{5}-\frac{81}{2}a^{4}+\frac{203}{6}a^{3}+\frac{2005}{6}a^{2}-615a+209$
| 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: | \( 211240.508114 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 211240.508114 \cdot 1}{2\cdot\sqrt{58280536320561}}\cr\approx \mathstrut & 1.77090558505 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^7 - 3*x^6 - 36*x^5 + 143*x^4 + 204*x^3 - 1476*x^2 + 1800*x - 432)
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^7 - 3*x^6 - 36*x^5 + 143*x^4 + 204*x^3 - 1476*x^2 + 1800*x - 432, 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^7 - 3*x^6 - 36*x^5 + 143*x^4 + 204*x^3 - 1476*x^2 + 1800*x - 432);
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^7 - 3*x^6 - 36*x^5 + 143*x^4 + 204*x^3 - 1476*x^2 + 1800*x - 432);
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 21 |
| The 5 conjugacy class representatives for $C_7:C_3$ |
| Character table for $C_7:C_3$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 21 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
|
\(307\)
| $\Q_{307}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $3$ | $3$ | $1$ | $2$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.2763.3t1.a.a | $1$ | $ 3^{2} \cdot 307 $ | 3.3.7634169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.2763.3t1.a.b | $1$ | $ 3^{2} \cdot 307 $ | 3.3.7634169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| * | 3.7634169.7t3.a.a | $3$ | $ 3^{4} \cdot 307^{2}$ | 7.7.58280536320561.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ |
| * | 3.7634169.7t3.a.b | $3$ | $ 3^{4} \cdot 307^{2}$ | 7.7.58280536320561.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.