Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^14 + 35*x^7 - 7)
gp: K = bnfinit(y^21 - 21*y^14 + 35*y^7 - 7, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^14 + 35*x^7 - 7);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 21*x^14 + 35*x^7 - 7)
\( x^{21} - 21x^{14} + 35x^{7} - 7 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-44567640326363195900190045974568007\)
\(\medspace = -\,7^{41}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(44.66\) | 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: | $7^{617/294}\approx 59.36850199103005$ | ||
| Ramified primes: |
\(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4}a^{7}+\frac{1}{4}$, $\frac{1}{4}a^{8}+\frac{1}{4}a$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{3}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{12}+\frac{1}{4}a^{5}$, $\frac{1}{4}a^{13}+\frac{1}{4}a^{6}$, $\frac{1}{16}a^{14}-\frac{1}{8}a^{7}-\frac{3}{16}$, $\frac{1}{16}a^{15}-\frac{1}{8}a^{8}-\frac{3}{16}a$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{9}-\frac{3}{16}a^{2}$, $\frac{1}{16}a^{17}-\frac{1}{8}a^{10}-\frac{3}{16}a^{3}$, $\frac{1}{16}a^{18}-\frac{1}{8}a^{11}-\frac{3}{16}a^{4}$, $\frac{1}{16}a^{19}-\frac{1}{8}a^{12}-\frac{3}{16}a^{5}$, $\frac{1}{16}a^{20}-\frac{1}{8}a^{13}-\frac{3}{16}a^{6}$
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 $1$ (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: |
$\frac{1}{16}a^{14}-\frac{9}{8}a^{7}+\frac{13}{16}$, $\frac{1}{16}a^{14}-\frac{9}{8}a^{7}-\frac{3}{16}$, $\frac{1}{8}a^{16}+\frac{1}{16}a^{15}-\frac{5}{2}a^{9}-\frac{11}{8}a^{8}+\frac{19}{8}a^{2}+\frac{41}{16}a+1$, $\frac{5}{16}a^{20}+\frac{1}{4}a^{18}-\frac{3}{16}a^{16}-\frac{3}{16}a^{15}-\frac{1}{8}a^{14}-\frac{51}{8}a^{13}+\frac{1}{4}a^{12}-5a^{11}+\frac{29}{8}a^{9}+\frac{29}{8}a^{8}+\frac{5}{2}a^{7}+\frac{117}{16}a^{6}-\frac{19}{4}a^{5}+\frac{15}{4}a^{4}-\frac{19}{16}a^{2}-\frac{3}{16}a-\frac{11}{8}$, $\frac{1}{4}a^{19}+\frac{3}{8}a^{18}+\frac{1}{8}a^{17}-\frac{1}{8}a^{16}-\frac{5}{16}a^{15}-\frac{1}{4}a^{14}-5a^{12}-\frac{15}{2}a^{11}-\frac{5}{2}a^{10}+\frac{5}{2}a^{9}+\frac{49}{8}a^{8}+5a^{7}+\frac{15}{4}a^{5}+\frac{49}{8}a^{4}+\frac{11}{8}a^{3}-\frac{11}{8}a^{2}-\frac{57}{16}a-\frac{11}{4}$, $\frac{3}{16}a^{18}-\frac{3}{8}a^{17}+\frac{3}{8}a^{16}-\frac{3}{16}a^{15}+\frac{1}{16}a^{14}-\frac{31}{8}a^{11}+\frac{31}{4}a^{10}-\frac{31}{4}a^{9}+\frac{31}{8}a^{8}-\frac{11}{8}a^{7}+\frac{79}{16}a^{4}-\frac{87}{8}a^{3}+\frac{87}{8}a^{2}-\frac{95}{16}a+\frac{25}{16}$, $\frac{3}{8}a^{20}-\frac{3}{16}a^{19}-\frac{1}{8}a^{18}+\frac{1}{8}a^{16}+\frac{1}{8}a^{14}-\frac{31}{4}a^{13}+\frac{31}{8}a^{12}+\frac{5}{2}a^{11}-\frac{5}{2}a^{9}+\frac{1}{4}a^{8}-\frac{9}{4}a^{7}+\frac{87}{8}a^{6}-\frac{79}{16}a^{5}-\frac{11}{8}a^{4}+a^{3}+\frac{27}{8}a^{2}-\frac{11}{4}a+\frac{5}{8}$, $\frac{1}{4}a^{20}-\frac{5}{16}a^{19}+\frac{1}{4}a^{18}-\frac{3}{16}a^{16}+\frac{3}{8}a^{15}-\frac{1}{2}a^{14}-\frac{19}{4}a^{13}+\frac{49}{8}a^{12}-5a^{11}+\frac{29}{8}a^{9}-\frac{29}{4}a^{8}+\frac{39}{4}a^{7}-a^{6}-\frac{41}{16}a^{5}+\frac{15}{4}a^{4}-\frac{3}{16}a^{2}+\frac{11}{8}a-\frac{11}{4}$, $\frac{7}{16}a^{20}+\frac{1}{4}a^{19}+\frac{1}{16}a^{18}+\frac{3}{16}a^{17}+\frac{1}{8}a^{16}+\frac{5}{16}a^{15}+\frac{3}{16}a^{14}-\frac{73}{8}a^{13}-\frac{21}{4}a^{12}-\frac{11}{8}a^{11}-\frac{31}{8}a^{10}-\frac{11}{4}a^{9}-\frac{49}{8}a^{8}-\frac{29}{8}a^{7}+\frac{215}{16}a^{6}+\frac{17}{2}a^{5}+\frac{73}{16}a^{4}+\frac{79}{16}a^{3}+\frac{33}{8}a^{2}+\frac{89}{16}a+\frac{67}{16}$, $\frac{1}{4}a^{20}+\frac{1}{16}a^{19}-\frac{1}{4}a^{18}-\frac{1}{16}a^{17}+\frac{1}{4}a^{16}+\frac{1}{16}a^{15}-\frac{1}{8}a^{14}-\frac{21}{4}a^{13}-\frac{11}{8}a^{12}+5a^{11}+\frac{11}{8}a^{10}-\frac{19}{4}a^{9}-\frac{9}{8}a^{8}+\frac{9}{4}a^{7}+\frac{17}{2}a^{6}+\frac{57}{16}a^{5}-\frac{15}{4}a^{4}-\frac{57}{16}a^{3}-\frac{3}{16}a+\frac{3}{8}$, $\frac{1}{2}a^{20}-\frac{1}{4}a^{19}+\frac{1}{2}a^{18}-\frac{9}{8}a^{17}+\frac{21}{16}a^{16}-\frac{7}{8}a^{15}+\frac{5}{16}a^{14}-\frac{41}{4}a^{13}+\frac{9}{2}a^{12}-\frac{37}{4}a^{11}+22a^{10}-\frac{207}{8}a^{9}+17a^{8}-\frac{45}{8}a^{7}+\frac{49}{4}a^{6}+\frac{23}{4}a^{5}-\frac{27}{4}a^{4}-\frac{55}{8}a^{3}+\frac{189}{16}a^{2}-\frac{41}{8}a-\frac{31}{16}$
| 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: | \( 2078079445.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 2078079445.0 \cdot 1}{2\cdot\sqrt{44567640326363195900190045974568007}}\cr\approx \mathstrut & 0.60094002799 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^14 + 35*x^7 - 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^21 - 21*x^14 + 35*x^7 - 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^21 - 21*x^14 + 35*x^7 - 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^21 - 21*x^14 + 35*x^7 - 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
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 294 |
| The 14 conjugacy class representatives for $C_7:F_7$ |
| Character table for $C_7:F_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
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 21 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| 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 | ${\href{/padicField/2.3.0.1}{3} }^{7}$ | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| Deg $21$ | $21$ | $1$ | $41$ |