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 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728)
gp: K = bnfinit(y^14 - 7*y^13 + 35*y^12 - 119*y^11 + 329*y^10 - 721*y^9 + 1337*y^8 + 355*y^7 - 5726*y^6 - 78484*y^5 + 212632*y^4 + 40096*y^3 - 276080*y^2 + 75152*y + 1455728, 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 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728);
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 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728)
\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} + 355 x^{7} + \cdots + 1455728 \)
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: |
\(-2523048836405617863000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{12}\cdot 5^{12}\cdot 7^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(148.45\) | 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: | $2^{6/7}3^{6/7}5^{6/7}7^{47/42}\approx 162.85791940621738$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{14}a^{7}-\frac{1}{2}a^{3}+\frac{3}{7}$, $\frac{1}{140}a^{8}-\frac{1}{35}a^{7}+\frac{1}{10}a^{6}-\frac{1}{5}a^{5}-\frac{3}{20}a^{4}-\frac{2}{5}a^{3}-\frac{1}{10}a^{2}+\frac{12}{35}a-\frac{6}{35}$, $\frac{1}{140}a^{9}-\frac{1}{70}a^{7}+\frac{1}{5}a^{6}+\frac{1}{20}a^{5}+\frac{3}{10}a^{3}-\frac{2}{35}a^{2}+\frac{1}{5}a+\frac{11}{35}$, $\frac{1}{140}a^{10}-\frac{1}{4}a^{6}+\frac{1}{10}a^{5}-\frac{5}{14}a^{3}-\frac{1}{2}a^{2}-\frac{1}{5}$, $\frac{1}{980}a^{11}+\frac{3}{980}a^{10}-\frac{1}{980}a^{9}+\frac{3}{980}a^{8}+\frac{3}{196}a^{7}-\frac{11}{140}a^{6}+\frac{3}{140}a^{5}-\frac{183}{980}a^{4}+\frac{17}{49}a^{3}-\frac{157}{490}a^{2}-\frac{13}{245}a+\frac{22}{49}$, $\frac{1}{1960}a^{12}-\frac{3}{1960}a^{10}+\frac{3}{980}a^{9}-\frac{1}{1960}a^{8}-\frac{3}{245}a^{7}+\frac{57}{280}a^{6}-\frac{221}{980}a^{5}+\frac{1}{35}a^{4}+\frac{167}{490}a^{3}-\frac{243}{490}a^{2}-\frac{18}{49}a-\frac{116}{245}$, $\frac{1}{18\!\cdots\!40}a^{13}-\frac{33\!\cdots\!51}{18\!\cdots\!40}a^{12}+\frac{81\!\cdots\!19}{18\!\cdots\!40}a^{11}+\frac{97\!\cdots\!75}{37\!\cdots\!08}a^{10}-\frac{22\!\cdots\!09}{18\!\cdots\!40}a^{9}-\frac{33\!\cdots\!99}{18\!\cdots\!40}a^{8}+\frac{69\!\cdots\!21}{18\!\cdots\!40}a^{7}+\frac{43\!\cdots\!77}{18\!\cdots\!40}a^{6}-\frac{58\!\cdots\!71}{46\!\cdots\!10}a^{5}-\frac{20\!\cdots\!21}{92\!\cdots\!20}a^{4}+\frac{18\!\cdots\!04}{23\!\cdots\!05}a^{3}+\frac{60\!\cdots\!73}{46\!\cdots\!10}a^{2}+\frac{22\!\cdots\!04}{23\!\cdots\!05}a-\frac{97\!\cdots\!38}{23\!\cdots\!05}$
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
$C_{7}$, which has order $7$ (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: |
$\frac{71\!\cdots\!17}{23\!\cdots\!05}a^{13}-\frac{35\!\cdots\!77}{23\!\cdots\!05}a^{12}+\frac{61\!\cdots\!27}{92\!\cdots\!20}a^{11}-\frac{79\!\cdots\!71}{46\!\cdots\!10}a^{10}+\frac{34\!\cdots\!89}{92\!\cdots\!20}a^{9}-\frac{49\!\cdots\!31}{92\!\cdots\!20}a^{8}+\frac{39\!\cdots\!37}{92\!\cdots\!20}a^{7}+\frac{16\!\cdots\!62}{23\!\cdots\!05}a^{6}-\frac{12\!\cdots\!81}{92\!\cdots\!20}a^{5}-\frac{25\!\cdots\!47}{92\!\cdots\!20}a^{4}+\frac{40\!\cdots\!53}{23\!\cdots\!05}a^{3}+\frac{32\!\cdots\!36}{23\!\cdots\!05}a^{2}+\frac{13\!\cdots\!03}{23\!\cdots\!05}a-\frac{28\!\cdots\!49}{23\!\cdots\!05}$, $\frac{45\!\cdots\!39}{92\!\cdots\!20}a^{13}-\frac{46\!\cdots\!32}{23\!\cdots\!05}a^{12}+\frac{10\!\cdots\!31}{92\!\cdots\!20}a^{11}-\frac{11\!\cdots\!51}{46\!\cdots\!10}a^{10}+\frac{42\!\cdots\!19}{46\!\cdots\!10}a^{9}-\frac{22\!\cdots\!07}{23\!\cdots\!05}a^{8}+\frac{34\!\cdots\!43}{92\!\cdots\!20}a^{7}+\frac{64\!\cdots\!11}{46\!\cdots\!10}a^{6}+\frac{15\!\cdots\!09}{18\!\cdots\!04}a^{5}-\frac{81\!\cdots\!74}{23\!\cdots\!05}a^{4}+\frac{87\!\cdots\!87}{23\!\cdots\!05}a^{3}+\frac{38\!\cdots\!89}{23\!\cdots\!05}a^{2}-\frac{24\!\cdots\!09}{23\!\cdots\!05}a-\frac{45\!\cdots\!69}{23\!\cdots\!05}$, $\frac{63\!\cdots\!49}{92\!\cdots\!20}a^{13}-\frac{48\!\cdots\!69}{92\!\cdots\!20}a^{12}+\frac{23\!\cdots\!21}{92\!\cdots\!20}a^{11}-\frac{38\!\cdots\!34}{46\!\cdots\!01}a^{10}+\frac{94\!\cdots\!89}{46\!\cdots\!10}a^{9}-\frac{65\!\cdots\!65}{18\!\cdots\!04}a^{8}+\frac{22\!\cdots\!37}{92\!\cdots\!20}a^{7}+\frac{50\!\cdots\!36}{23\!\cdots\!05}a^{6}-\frac{94\!\cdots\!31}{92\!\cdots\!20}a^{5}-\frac{20\!\cdots\!97}{46\!\cdots\!10}a^{4}+\frac{15\!\cdots\!31}{92\!\cdots\!02}a^{3}+\frac{79\!\cdots\!13}{23\!\cdots\!05}a^{2}-\frac{13\!\cdots\!98}{23\!\cdots\!05}a+\frac{23\!\cdots\!59}{23\!\cdots\!05}$, $\frac{29\!\cdots\!12}{46\!\cdots\!01}a^{13}-\frac{10\!\cdots\!69}{46\!\cdots\!10}a^{12}+\frac{13\!\cdots\!23}{92\!\cdots\!20}a^{11}-\frac{24\!\cdots\!01}{92\!\cdots\!20}a^{10}+\frac{56\!\cdots\!03}{46\!\cdots\!10}a^{9}-\frac{27\!\cdots\!83}{46\!\cdots\!10}a^{8}+\frac{12\!\cdots\!13}{18\!\cdots\!04}a^{7}+\frac{21\!\cdots\!81}{92\!\cdots\!20}a^{6}+\frac{19\!\cdots\!63}{46\!\cdots\!10}a^{5}-\frac{82\!\cdots\!76}{23\!\cdots\!05}a^{4}+\frac{16\!\cdots\!22}{23\!\cdots\!05}a^{3}+\frac{50\!\cdots\!82}{46\!\cdots\!01}a^{2}-\frac{32\!\cdots\!46}{23\!\cdots\!05}a-\frac{72\!\cdots\!91}{23\!\cdots\!05}$, $\frac{23\!\cdots\!13}{18\!\cdots\!40}a^{13}-\frac{48\!\cdots\!57}{37\!\cdots\!08}a^{12}+\frac{12\!\cdots\!33}{18\!\cdots\!40}a^{11}-\frac{30\!\cdots\!01}{18\!\cdots\!40}a^{10}+\frac{69\!\cdots\!13}{18\!\cdots\!40}a^{9}+\frac{20\!\cdots\!41}{18\!\cdots\!40}a^{8}-\frac{11\!\cdots\!17}{18\!\cdots\!40}a^{7}-\frac{24\!\cdots\!03}{18\!\cdots\!40}a^{6}+\frac{58\!\cdots\!65}{18\!\cdots\!04}a^{5}+\frac{25\!\cdots\!77}{23\!\cdots\!05}a^{4}-\frac{17\!\cdots\!99}{23\!\cdots\!05}a^{3}-\frac{41\!\cdots\!01}{92\!\cdots\!02}a^{2}+\frac{46\!\cdots\!84}{23\!\cdots\!05}a+\frac{11\!\cdots\!73}{23\!\cdots\!05}$, $\frac{24\!\cdots\!36}{33\!\cdots\!15}a^{13}-\frac{15\!\cdots\!61}{46\!\cdots\!10}a^{12}+\frac{16\!\cdots\!57}{92\!\cdots\!20}a^{11}-\frac{78\!\cdots\!39}{18\!\cdots\!04}a^{10}+\frac{61\!\cdots\!29}{46\!\cdots\!10}a^{9}-\frac{25\!\cdots\!73}{13\!\cdots\!60}a^{8}+\frac{47\!\cdots\!23}{92\!\cdots\!20}a^{7}+\frac{56\!\cdots\!93}{37\!\cdots\!96}a^{6}-\frac{90\!\cdots\!47}{46\!\cdots\!01}a^{5}-\frac{55\!\cdots\!27}{92\!\cdots\!20}a^{4}-\frac{46\!\cdots\!37}{92\!\cdots\!02}a^{3}+\frac{72\!\cdots\!57}{46\!\cdots\!10}a^{2}-\frac{54\!\cdots\!32}{33\!\cdots\!15}a-\frac{92\!\cdots\!49}{23\!\cdots\!05}$
| 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: | \( 1187793284.646849 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 1187793284.646849 \cdot 7}{2\cdot\sqrt{2523048836405617863000000000000}}\cr\approx \mathstrut & 1.01182442769803 \end{aligned}\] (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 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728)
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 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728, 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 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728);
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 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728);
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
| \(\Q(\sqrt{-7}) \), 7.1.600362847000000.17 |
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: | 7.1.600362847000000.17 |
| Degree 21 sibling: | deg 21 |
| Minimal sibling: | 7.1.600362847000000.17 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
|
\(3\)
| 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
|
\(5\)
| 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 | $x^{14} + 7 x^{3} + 7 x^{2} + 7$ | $14$ | $1$ | $15$ | $F_7$ | $[7/6]_{6}$ |