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 + 375*x^7 - 5796*x^6 - 79114*x^5 + 214382*x^4 + 40446*x^3 - 278390*x^2 + 75782*x + 1479958)
gp: K = bnfinit(y^14 - 7*y^13 + 35*y^12 - 119*y^11 + 329*y^10 - 721*y^9 + 1337*y^8 + 375*y^7 - 5796*y^6 - 79114*y^5 + 214382*y^4 + 40446*y^3 - 278390*y^2 + 75782*y + 1479958, 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 + 375*x^7 - 5796*x^6 - 79114*x^5 + 214382*x^4 + 40446*x^3 - 278390*x^2 + 75782*x + 1479958);
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 + 375*x^7 - 5796*x^6 - 79114*x^5 + 214382*x^4 + 40446*x^3 - 278390*x^2 + 75782*x + 1479958)
\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} + 375 x^{7} + \cdots + 1479958 \)
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: |
\(-14899881763033509191236903000000000000\)
\(\medspace = -\,2^{12}\cdot 5^{12}\cdot 7^{15}\cdot 11^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(452.09\) | 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: | $2^{6/7}5^{6/7}7^{47/42}11^{6/7}\approx 495.9867882910352$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}+\frac{2}{7}$, $\frac{1}{77}a^{8}-\frac{4}{77}a^{7}+\frac{2}{11}a^{6}-\frac{4}{11}a^{5}-\frac{4}{11}a^{4}+\frac{3}{11}a^{3}-\frac{3}{11}a^{2}+\frac{23}{77}a+\frac{27}{77}$, $\frac{1}{77}a^{9}-\frac{2}{77}a^{7}+\frac{4}{11}a^{6}+\frac{2}{11}a^{5}-\frac{2}{11}a^{4}-\frac{2}{11}a^{3}+\frac{16}{77}a^{2}-\frac{5}{11}a+\frac{31}{77}$, $\frac{1}{77}a^{10}-\frac{2}{77}a^{7}-\frac{5}{11}a^{6}+\frac{1}{11}a^{5}+\frac{1}{11}a^{4}-\frac{19}{77}a^{3}+\frac{10}{77}$, $\frac{1}{539}a^{11}+\frac{1}{539}a^{10}+\frac{3}{539}a^{9}-\frac{3}{539}a^{8}+\frac{16}{539}a^{7}-\frac{27}{77}a^{6}+\frac{12}{77}a^{5}-\frac{103}{539}a^{4}-\frac{236}{539}a^{3}-\frac{85}{539}a^{2}+\frac{267}{539}a+\frac{263}{539}$, $\frac{1}{539}a^{12}+\frac{2}{539}a^{10}+\frac{1}{539}a^{9}-\frac{2}{539}a^{8}+\frac{19}{539}a^{7}+\frac{25}{77}a^{6}-\frac{40}{539}a^{5}-\frac{26}{77}a^{4}+\frac{151}{539}a^{3}-\frac{173}{539}a^{2}-\frac{193}{539}a+\frac{234}{539}$, $\frac{1}{52\!\cdots\!11}a^{13}+\frac{20\!\cdots\!42}{52\!\cdots\!11}a^{12}+\frac{20\!\cdots\!40}{40\!\cdots\!47}a^{11}-\frac{27\!\cdots\!58}{47\!\cdots\!01}a^{10}-\frac{12\!\cdots\!87}{52\!\cdots\!11}a^{9}-\frac{14\!\cdots\!40}{52\!\cdots\!11}a^{8}-\frac{31\!\cdots\!70}{74\!\cdots\!73}a^{7}-\frac{14\!\cdots\!14}{52\!\cdots\!11}a^{6}+\frac{81\!\cdots\!45}{40\!\cdots\!47}a^{5}+\frac{19\!\cdots\!03}{52\!\cdots\!11}a^{4}+\frac{10\!\cdots\!34}{40\!\cdots\!47}a^{3}+\frac{15\!\cdots\!02}{52\!\cdots\!11}a^{2}+\frac{10\!\cdots\!49}{52\!\cdots\!11}a-\frac{33\!\cdots\!97}{10\!\cdots\!39}$
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
$C_{7}\times C_{7}$, which has order $49$ (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: |
$\frac{57\!\cdots\!33}{74\!\cdots\!73}a^{13}-\frac{30\!\cdots\!51}{52\!\cdots\!11}a^{12}+\frac{83\!\cdots\!37}{40\!\cdots\!47}a^{11}-\frac{55\!\cdots\!48}{47\!\cdots\!01}a^{10}-\frac{89\!\cdots\!94}{52\!\cdots\!11}a^{9}+\frac{26\!\cdots\!14}{52\!\cdots\!11}a^{8}+\frac{51\!\cdots\!09}{47\!\cdots\!01}a^{7}-\frac{76\!\cdots\!89}{10\!\cdots\!39}a^{6}-\frac{48\!\cdots\!52}{40\!\cdots\!47}a^{5}+\frac{15\!\cdots\!82}{52\!\cdots\!11}a^{4}-\frac{30\!\cdots\!75}{40\!\cdots\!47}a^{3}-\frac{25\!\cdots\!30}{52\!\cdots\!11}a^{2}+\frac{33\!\cdots\!56}{52\!\cdots\!11}a+\frac{83\!\cdots\!07}{52\!\cdots\!11}$, $\frac{12\!\cdots\!80}{52\!\cdots\!11}a^{13}-\frac{65\!\cdots\!69}{52\!\cdots\!11}a^{12}+\frac{16\!\cdots\!60}{40\!\cdots\!47}a^{11}-\frac{99\!\cdots\!02}{52\!\cdots\!11}a^{10}+\frac{10\!\cdots\!96}{52\!\cdots\!11}a^{9}-\frac{69\!\cdots\!85}{74\!\cdots\!73}a^{8}+\frac{19\!\cdots\!49}{10\!\cdots\!39}a^{7}+\frac{64\!\cdots\!71}{52\!\cdots\!11}a^{6}+\frac{11\!\cdots\!96}{40\!\cdots\!47}a^{5}-\frac{74\!\cdots\!31}{52\!\cdots\!11}a^{4}+\frac{59\!\cdots\!85}{40\!\cdots\!47}a^{3}+\frac{93\!\cdots\!95}{52\!\cdots\!11}a^{2}-\frac{15\!\cdots\!01}{74\!\cdots\!73}a-\frac{73\!\cdots\!27}{74\!\cdots\!73}$, $\frac{25\!\cdots\!09}{52\!\cdots\!11}a^{13}-\frac{24\!\cdots\!29}{74\!\cdots\!73}a^{12}+\frac{60\!\cdots\!82}{40\!\cdots\!47}a^{11}-\frac{25\!\cdots\!26}{52\!\cdots\!11}a^{10}+\frac{64\!\cdots\!50}{52\!\cdots\!11}a^{9}-\frac{13\!\cdots\!04}{52\!\cdots\!11}a^{8}+\frac{22\!\cdots\!25}{52\!\cdots\!11}a^{7}+\frac{29\!\cdots\!33}{52\!\cdots\!11}a^{6}-\frac{18\!\cdots\!40}{57\!\cdots\!21}a^{5}-\frac{21\!\cdots\!52}{52\!\cdots\!11}a^{4}+\frac{40\!\cdots\!46}{40\!\cdots\!47}a^{3}+\frac{81\!\cdots\!04}{52\!\cdots\!11}a^{2}-\frac{73\!\cdots\!44}{52\!\cdots\!11}a-\frac{25\!\cdots\!79}{52\!\cdots\!11}$, $\frac{34\!\cdots\!74}{52\!\cdots\!11}a^{13}-\frac{43\!\cdots\!64}{74\!\cdots\!73}a^{12}+\frac{13\!\cdots\!37}{40\!\cdots\!47}a^{11}-\frac{77\!\cdots\!37}{52\!\cdots\!11}a^{10}+\frac{26\!\cdots\!89}{52\!\cdots\!11}a^{9}-\frac{10\!\cdots\!03}{74\!\cdots\!73}a^{8}+\frac{14\!\cdots\!11}{52\!\cdots\!11}a^{7}-\frac{92\!\cdots\!39}{52\!\cdots\!11}a^{6}-\frac{26\!\cdots\!73}{57\!\cdots\!21}a^{5}-\frac{27\!\cdots\!97}{52\!\cdots\!11}a^{4}+\frac{10\!\cdots\!41}{40\!\cdots\!47}a^{3}-\frac{20\!\cdots\!61}{52\!\cdots\!11}a^{2}+\frac{12\!\cdots\!56}{74\!\cdots\!73}a+\frac{17\!\cdots\!73}{52\!\cdots\!11}$, $\frac{48\!\cdots\!05}{52\!\cdots\!11}a^{13}-\frac{14\!\cdots\!13}{74\!\cdots\!73}a^{12}+\frac{52\!\cdots\!10}{40\!\cdots\!47}a^{11}-\frac{15\!\cdots\!11}{52\!\cdots\!11}a^{10}-\frac{19\!\cdots\!07}{52\!\cdots\!11}a^{9}-\frac{29\!\cdots\!53}{74\!\cdots\!73}a^{8}-\frac{31\!\cdots\!91}{52\!\cdots\!11}a^{7}+\frac{57\!\cdots\!43}{52\!\cdots\!11}a^{6}+\frac{65\!\cdots\!62}{57\!\cdots\!21}a^{5}-\frac{68\!\cdots\!94}{52\!\cdots\!11}a^{4}-\frac{15\!\cdots\!14}{40\!\cdots\!47}a^{3}+\frac{22\!\cdots\!42}{52\!\cdots\!11}a^{2}+\frac{38\!\cdots\!15}{74\!\cdots\!73}a-\frac{39\!\cdots\!55}{52\!\cdots\!11}$, $\frac{24\!\cdots\!35}{52\!\cdots\!11}a^{13}-\frac{38\!\cdots\!53}{74\!\cdots\!73}a^{12}+\frac{69\!\cdots\!57}{40\!\cdots\!47}a^{11}-\frac{15\!\cdots\!45}{52\!\cdots\!11}a^{10}+\frac{10\!\cdots\!71}{74\!\cdots\!73}a^{9}-\frac{22\!\cdots\!94}{47\!\cdots\!01}a^{8}-\frac{35\!\cdots\!29}{52\!\cdots\!11}a^{7}+\frac{15\!\cdots\!08}{52\!\cdots\!11}a^{6}+\frac{82\!\cdots\!22}{57\!\cdots\!21}a^{5}-\frac{20\!\cdots\!91}{52\!\cdots\!11}a^{4}-\frac{24\!\cdots\!18}{40\!\cdots\!47}a^{3}+\frac{46\!\cdots\!90}{74\!\cdots\!73}a^{2}-\frac{21\!\cdots\!13}{52\!\cdots\!11}a-\frac{12\!\cdots\!91}{52\!\cdots\!11}$
| 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: | \( 628315297159.5366 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 628315297159.5366 \cdot 49}{2\cdot\sqrt{14899881763033509191236903000000000000}}\cr\approx \mathstrut & 1.54174086950575 \end{aligned}\] (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 + 375*x^7 - 5796*x^6 - 79114*x^5 + 214382*x^4 + 40446*x^3 - 278390*x^2 + 75782*x + 1479958)
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 + 375*x^7 - 5796*x^6 - 79114*x^5 + 214382*x^4 + 40446*x^3 - 278390*x^2 + 75782*x + 1479958, 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 + 375*x^7 - 5796*x^6 - 79114*x^5 + 214382*x^4 + 40446*x^3 - 278390*x^2 + 75782*x + 1479958);
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 + 375*x^7 - 5796*x^6 - 79114*x^5 + 214382*x^4 + 40446*x^3 - 278390*x^2 + 75782*x + 1479958);
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
| \(\Q(\sqrt{-7}) \), 7.1.1458956660623000000.11 |
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: | 7.1.1458956660623000000.11 |
| Degree 21 sibling: | deg 21 |
| Minimal sibling: | 7.1.1458956660623000000.11 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | R | R | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.1.0.1}{1} }^{14}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
|
\(5\)
| 5.14.12.1 | $x^{14} + 28 x^{13} + 350 x^{12} + 2576 x^{11} + 12404 x^{10} + 41104 x^{9} + 96152 x^{8} + 160650 x^{7} + 192444 x^{6} + 165676 x^{5} + 106232 x^{4} + 65016 x^{3} + 59920 x^{2} + 57232 x + 27193$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
|
\(7\)
| 7.14.15.5 | $x^{14} + 7 x^{3} + 7 x^{2} + 7$ | $14$ | $1$ | $15$ | $F_7$ | $[7/6]_{6}$ |
|
\(11\)
| 11.7.6.1 | $x^{7} + 11$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| 11.7.6.1 | $x^{7} + 11$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |