Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 - 1909*x^7 + 2198*x^6 - 7168*x^5 + 14532*x^4 + 476*x^3 - 14588*x^2 + 3836*x + 5636)
gp: K = bnfinit(y^14 - 7*y^13 + 35*y^12 - 119*y^11 + 329*y^10 - 721*y^9 + 1337*y^8 - 1909*y^7 + 2198*y^6 - 7168*y^5 + 14532*y^4 + 476*y^3 - 14588*y^2 + 3836*y + 5636, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 - 1909*x^7 + 2198*x^6 - 7168*x^5 + 14532*x^4 + 476*x^3 - 14588*x^2 + 3836*x + 5636);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 - 1909*x^7 + 2198*x^6 - 7168*x^5 + 14532*x^4 + 476*x^3 - 14588*x^2 + 3836*x + 5636)
\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} - 1909 x^{7} + \cdots + 5636 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-4718733437914712722630340608\)
\(\medspace = -\,2^{12}\cdot 7^{11}\cdot 17^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(94.78\) | 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: | $2^{6/7}7^{5/6}17^{6/7}\approx 103.9787473926292$ | ||
| Ramified primes: |
\(2\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{3}+\frac{1}{7}a^{2}-\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{7}+\frac{1}{7}$, $\frac{1}{14}a^{8}-\frac{1}{2}a^{4}-\frac{3}{7}a$, $\frac{1}{14}a^{9}-\frac{1}{2}a^{5}-\frac{3}{7}a^{2}$, $\frac{1}{98}a^{10}+\frac{3}{98}a^{9}+\frac{3}{98}a^{8}-\frac{3}{49}a^{7}-\frac{1}{14}a^{6}-\frac{5}{14}a^{5}-\frac{3}{14}a^{4}-\frac{10}{49}a^{3}-\frac{16}{49}a^{2}-\frac{2}{49}a-\frac{3}{49}$, $\frac{1}{98}a^{11}+\frac{1}{98}a^{9}-\frac{1}{98}a^{8}-\frac{3}{98}a^{7}+\frac{3}{14}a^{5}-\frac{41}{98}a^{4}+\frac{1}{7}a^{3}-\frac{17}{49}a^{2}+\frac{3}{49}a+\frac{9}{49}$, $\frac{1}{98}a^{12}+\frac{3}{98}a^{9}+\frac{1}{98}a^{8}+\frac{3}{49}a^{7}-\frac{27}{98}a^{5}-\frac{3}{7}a^{4}+\frac{1}{7}a^{3}-\frac{16}{49}a^{2}+\frac{4}{49}a-\frac{11}{49}$, $\frac{1}{81\!\cdots\!22}a^{13}+\frac{36271612257047}{81\!\cdots\!22}a^{12}+\frac{5764081374655}{40\!\cdots\!61}a^{11}-\frac{8768135047289}{81\!\cdots\!22}a^{10}+\frac{51661378404189}{81\!\cdots\!22}a^{9}+\frac{2874527542617}{11\!\cdots\!46}a^{8}+\frac{260122636554969}{40\!\cdots\!61}a^{7}+\frac{67934632067429}{81\!\cdots\!22}a^{6}-\frac{19\!\cdots\!33}{40\!\cdots\!61}a^{5}+\frac{20\!\cdots\!71}{40\!\cdots\!61}a^{4}+\frac{142003408155680}{313426366797397}a^{3}+\frac{14\!\cdots\!96}{40\!\cdots\!61}a^{2}-\frac{143772893619850}{582077538338023}a-\frac{84243874861889}{40\!\cdots\!61}$
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: | $6$ | 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: |
$\frac{211833073030724}{40\!\cdots\!61}a^{13}-\frac{16\!\cdots\!84}{40\!\cdots\!61}a^{12}+\frac{16\!\cdots\!29}{81\!\cdots\!22}a^{11}-\frac{30\!\cdots\!95}{40\!\cdots\!61}a^{10}+\frac{17\!\cdots\!77}{81\!\cdots\!22}a^{9}-\frac{42\!\cdots\!33}{81\!\cdots\!22}a^{8}+\frac{83\!\cdots\!37}{81\!\cdots\!22}a^{7}-\frac{67\!\cdots\!62}{40\!\cdots\!61}a^{6}+\frac{17\!\cdots\!35}{81\!\cdots\!22}a^{5}-\frac{41\!\cdots\!55}{81\!\cdots\!22}a^{4}+\frac{33\!\cdots\!14}{313426366797397}a^{3}-\frac{27\!\cdots\!85}{40\!\cdots\!61}a^{2}-\frac{14\!\cdots\!83}{40\!\cdots\!61}a+\frac{18\!\cdots\!91}{40\!\cdots\!61}$, $\frac{260350299143325}{81\!\cdots\!22}a^{13}-\frac{803433127084979}{40\!\cdots\!61}a^{12}+\frac{77\!\cdots\!75}{81\!\cdots\!22}a^{11}-\frac{12\!\cdots\!89}{40\!\cdots\!61}a^{10}+\frac{31\!\cdots\!22}{40\!\cdots\!61}a^{9}-\frac{13\!\cdots\!55}{81\!\cdots\!22}a^{8}+\frac{22\!\cdots\!47}{81\!\cdots\!22}a^{7}-\frac{14\!\cdots\!11}{40\!\cdots\!61}a^{6}+\frac{29\!\cdots\!15}{81\!\cdots\!22}a^{5}-\frac{15\!\cdots\!45}{81\!\cdots\!22}a^{4}+\frac{93\!\cdots\!76}{313426366797397}a^{3}+\frac{12\!\cdots\!54}{40\!\cdots\!61}a^{2}-\frac{95\!\cdots\!50}{40\!\cdots\!61}a-\frac{68\!\cdots\!19}{40\!\cdots\!61}$, $\frac{55676510073139}{81\!\cdots\!22}a^{13}-\frac{14\!\cdots\!52}{40\!\cdots\!61}a^{12}+\frac{25\!\cdots\!91}{81\!\cdots\!22}a^{11}-\frac{73\!\cdots\!01}{40\!\cdots\!61}a^{10}+\frac{30\!\cdots\!93}{40\!\cdots\!61}a^{9}-\frac{19\!\cdots\!66}{83153934048289}a^{8}+\frac{46\!\cdots\!89}{81\!\cdots\!22}a^{7}-\frac{45\!\cdots\!46}{40\!\cdots\!61}a^{6}+\frac{12\!\cdots\!55}{81\!\cdots\!22}a^{5}-\frac{30\!\cdots\!30}{40\!\cdots\!61}a^{4}-\frac{33\!\cdots\!99}{313426366797397}a^{3}+\frac{53\!\cdots\!47}{40\!\cdots\!61}a^{2}+\frac{26\!\cdots\!29}{582077538338023}a-\frac{18\!\cdots\!85}{40\!\cdots\!61}$, $\frac{13\!\cdots\!66}{40\!\cdots\!61}a^{13}-\frac{15\!\cdots\!85}{81\!\cdots\!22}a^{12}+\frac{37\!\cdots\!22}{40\!\cdots\!61}a^{11}-\frac{23\!\cdots\!71}{81\!\cdots\!22}a^{10}+\frac{30\!\cdots\!54}{40\!\cdots\!61}a^{9}-\frac{12\!\cdots\!45}{81\!\cdots\!22}a^{8}+\frac{10\!\cdots\!79}{40\!\cdots\!61}a^{7}-\frac{25\!\cdots\!03}{81\!\cdots\!22}a^{6}+\frac{13\!\cdots\!62}{40\!\cdots\!61}a^{5}-\frac{79\!\cdots\!22}{40\!\cdots\!61}a^{4}+\frac{81\!\cdots\!17}{313426366797397}a^{3}+\frac{14\!\cdots\!21}{40\!\cdots\!61}a^{2}-\frac{54\!\cdots\!93}{40\!\cdots\!61}a-\frac{52\!\cdots\!19}{40\!\cdots\!61}$, $\frac{88\!\cdots\!57}{81\!\cdots\!22}a^{13}-\frac{12\!\cdots\!37}{166307868096578}a^{12}+\frac{67\!\cdots\!82}{40\!\cdots\!61}a^{11}-\frac{26\!\cdots\!38}{40\!\cdots\!61}a^{10}+\frac{24\!\cdots\!69}{81\!\cdots\!22}a^{9}+\frac{11\!\cdots\!59}{40\!\cdots\!61}a^{8}-\frac{24\!\cdots\!46}{40\!\cdots\!61}a^{7}+\frac{89\!\cdots\!95}{40\!\cdots\!61}a^{6}-\frac{29\!\cdots\!89}{582077538338023}a^{5}-\frac{44\!\cdots\!61}{81\!\cdots\!22}a^{4}-\frac{49\!\cdots\!21}{313426366797397}a^{3}+\frac{99\!\cdots\!04}{40\!\cdots\!61}a^{2}+\frac{73\!\cdots\!22}{40\!\cdots\!61}a-\frac{55\!\cdots\!89}{40\!\cdots\!61}$, $\frac{34\!\cdots\!62}{313426366797397}a^{13}-\frac{17\!\cdots\!17}{313426366797397}a^{12}+\frac{17\!\cdots\!35}{626852733594794}a^{11}-\frac{47\!\cdots\!15}{626852733594794}a^{10}+\frac{12\!\cdots\!59}{626852733594794}a^{9}-\frac{11\!\cdots\!52}{313426366797397}a^{8}+\frac{42\!\cdots\!73}{626852733594794}a^{7}-\frac{43\!\cdots\!89}{626852733594794}a^{6}+\frac{52\!\cdots\!15}{626852733594794}a^{5}-\frac{19\!\cdots\!54}{313426366797397}a^{4}+\frac{75\!\cdots\!49}{313426366797397}a^{3}+\frac{19\!\cdots\!84}{313426366797397}a^{2}-\frac{99\!\cdots\!15}{313426366797397}a-\frac{89\!\cdots\!23}{313426366797397}$
| 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: | \( 605619383.0164032 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 605619383.0164032 \cdot 1}{2\cdot\sqrt{4718733437914712722630340608}}\cr\approx \mathstrut & 1.70418260066651 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 - 1909*x^7 + 2198*x^6 - 7168*x^5 + 14532*x^4 + 476*x^3 - 14588*x^2 + 3836*x + 5636)
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^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 - 1909*x^7 + 2198*x^6 - 7168*x^5 + 14532*x^4 + 476*x^3 - 14588*x^2 + 3836*x + 5636, 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^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 - 1909*x^7 + 2198*x^6 - 7168*x^5 + 14532*x^4 + 476*x^3 - 14588*x^2 + 3836*x + 5636);
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^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 - 1909*x^7 + 2198*x^6 - 7168*x^5 + 14532*x^4 + 476*x^3 - 14588*x^2 + 3836*x + 5636);
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 42 |
| The 7 conjugacy class representatives for $F_7$ |
| Character table for $F_7$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 7.1.25963527819712.1 |
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 42 |
| Degree 7 sibling: | 7.1.25963527819712.1 |
| Degree 21 sibling: | deg 21 |
| Minimal sibling: | 7.1.25963527819712.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
|
\(17\)
| 17.14.12.1 | $x^{14} + 112 x^{13} + 5397 x^{12} + 145376 x^{11} + 2374589 x^{10} + 23755536 x^{9} + 138569137 x^{8} + 408357986 x^{7} + 415709315 x^{6} + 213889074 x^{5} + 66465343 x^{4} + 48952246 x^{3} + 353976063 x^{2} + 1858653398 x + 4197785820$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |