Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 31*x^12 + 56*x^11 + 314*x^10 - 584*x^9 - 1306*x^8 + 2870*x^7 + 1747*x^6 - 6530*x^5 + 1987*x^4 + 4874*x^3 - 4973*x^2 + 1816*x - 238)
gp: K = bnfinit(y^14 - 2*y^13 - 31*y^12 + 56*y^11 + 314*y^10 - 584*y^9 - 1306*y^8 + 2870*y^7 + 1747*y^6 - 6530*y^5 + 1987*y^4 + 4874*y^3 - 4973*y^2 + 1816*y - 238, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x^13 - 31*x^12 + 56*x^11 + 314*x^10 - 584*x^9 - 1306*x^8 + 2870*x^7 + 1747*x^6 - 6530*x^5 + 1987*x^4 + 4874*x^3 - 4973*x^2 + 1816*x - 238);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 2*x^13 - 31*x^12 + 56*x^11 + 314*x^10 - 584*x^9 - 1306*x^8 + 2870*x^7 + 1747*x^6 - 6530*x^5 + 1987*x^4 + 4874*x^3 - 4973*x^2 + 1816*x - 238)
\( x^{14} - 2 x^{13} - 31 x^{12} + 56 x^{11} + 314 x^{10} - 584 x^{9} - 1306 x^{8} + 2870 x^{7} + \cdots - 238 \)
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: | $[14, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(153645020121345627783168\)
\(\medspace = 2^{21}\cdot 3^{12}\cdot 13^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(45.31\) | 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: | $2^{3/2}3^{6/7}13^{5/6}\approx 61.48825892056607$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13}a^{8}+\frac{1}{13}a^{7}+\frac{4}{13}a^{6}+\frac{5}{13}a^{5}+\frac{6}{13}a^{4}+\frac{3}{13}a^{3}+\frac{1}{13}a^{2}+\frac{4}{13}$, $\frac{1}{13}a^{9}+\frac{3}{13}a^{7}+\frac{1}{13}a^{6}+\frac{1}{13}a^{5}-\frac{3}{13}a^{4}-\frac{2}{13}a^{3}-\frac{1}{13}a^{2}+\frac{4}{13}a-\frac{4}{13}$, $\frac{1}{13}a^{10}-\frac{2}{13}a^{7}+\frac{2}{13}a^{6}-\frac{5}{13}a^{5}+\frac{6}{13}a^{4}+\frac{3}{13}a^{3}+\frac{1}{13}a^{2}-\frac{4}{13}a+\frac{1}{13}$, $\frac{1}{13}a^{11}+\frac{4}{13}a^{7}+\frac{3}{13}a^{6}+\frac{3}{13}a^{5}+\frac{2}{13}a^{4}-\frac{6}{13}a^{3}-\frac{2}{13}a^{2}+\frac{1}{13}a-\frac{5}{13}$, $\frac{1}{13}a^{12}-\frac{1}{13}a^{7}-\frac{5}{13}a^{5}-\frac{4}{13}a^{4}-\frac{1}{13}a^{3}-\frac{3}{13}a^{2}-\frac{5}{13}a-\frac{3}{13}$, $\frac{1}{143}a^{13}-\frac{4}{143}a^{12}-\frac{1}{143}a^{11}+\frac{3}{143}a^{10}-\frac{1}{143}a^{8}-\frac{6}{143}a^{7}+\frac{1}{13}a^{6}-\frac{2}{143}a^{5}-\frac{47}{143}a^{4}+\frac{68}{143}a^{3}-\frac{14}{143}a^{2}-\frac{61}{143}a+\frac{46}{143}$
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: | $13$ | 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: |
$\frac{144}{143}a^{13}-\frac{180}{143}a^{12}-\frac{4599}{143}a^{11}+\frac{4623}{143}a^{10}+\frac{4424}{13}a^{9}-\frac{47829}{143}a^{8}-\frac{223449}{143}a^{7}+\frac{22512}{13}a^{6}+\frac{433167}{143}a^{5}-\frac{621129}{143}a^{4}-\frac{165372}{143}a^{3}+\frac{579642}{143}a^{2}-\frac{298359}{143}a+\frac{47907}{143}$, $\frac{361}{143}a^{13}-\frac{29}{11}a^{12}-\frac{11449}{143}a^{11}+\frac{9278}{143}a^{10}+\frac{10824}{13}a^{9}-\frac{97909}{143}a^{8}-\frac{534632}{143}a^{7}+\frac{48383}{13}a^{6}+\frac{1015777}{143}a^{5}-\frac{1385884}{143}a^{4}-\frac{381572}{143}a^{3}+\frac{1311151}{143}a^{2}-\frac{690634}{143}a+\frac{115925}{143}$, $\frac{98}{143}a^{13}+\frac{48}{143}a^{12}-\frac{3123}{143}a^{11}-\frac{2214}{143}a^{10}+\frac{2882}{13}a^{9}+\frac{20967}{143}a^{8}-\frac{141047}{143}a^{7}-\frac{5604}{13}a^{6}+\frac{303404}{143}a^{5}+\frac{30704}{143}a^{4}-\frac{291810}{143}a^{3}+\frac{70293}{143}a^{2}+\frac{69009}{143}a-\frac{24741}{143}$, $\frac{431}{143}a^{13}-\frac{635}{143}a^{12}-\frac{13708}{143}a^{11}+\frac{16891}{143}a^{10}+\frac{13147}{13}a^{9}-\frac{174704}{143}a^{8}-\frac{658373}{143}a^{7}+\frac{80279}{13}a^{6}+\frac{94653}{11}a^{5}-\frac{2145204}{143}a^{4}-\frac{294158}{143}a^{3}+\frac{148167}{11}a^{2}-\frac{1112090}{143}a+\frac{198213}{143}$, $\frac{214}{143}a^{13}-\frac{10}{11}a^{12}-\frac{6781}{143}a^{11}+\frac{2578}{143}a^{10}+\frac{6341}{13}a^{9}-\frac{29210}{143}a^{8}-\frac{310780}{143}a^{7}+\frac{17595}{13}a^{6}+\frac{608763}{143}a^{5}-\frac{45025}{11}a^{4}-\frac{330617}{143}a^{3}+\frac{611277}{143}a^{2}-\frac{258497}{143}a+\frac{35551}{143}$, $\frac{1}{13}a^{13}+\frac{2}{13}a^{12}-\frac{31}{13}a^{11}-5a^{10}+\frac{281}{13}a^{9}+\frac{534}{13}a^{8}-\frac{997}{13}a^{7}-\frac{1317}{13}a^{6}+\frac{1315}{13}a^{5}+\frac{8}{13}a^{4}+\frac{782}{13}a^{3}+\frac{1761}{13}a^{2}-\frac{3353}{13}a+\frac{1087}{13}$, $\frac{438}{143}a^{13}-\frac{531}{143}a^{12}-\frac{13836}{143}a^{11}+\frac{13700}{143}a^{10}+\frac{13025}{13}a^{9}-\frac{146144}{143}a^{8}-\frac{637075}{143}a^{7}+\frac{71085}{13}a^{6}+\frac{1166785}{143}a^{5}-\frac{1987144}{143}a^{4}-\frac{264675}{143}a^{3}+\frac{1822200}{143}a^{2}-\frac{1073786}{143}a+\frac{194333}{143}$, $\frac{57}{143}a^{13}-\frac{151}{143}a^{12}-\frac{1806}{143}a^{11}+\frac{4362}{143}a^{10}+\frac{1763}{13}a^{9}-\frac{44838}{143}a^{8}-\frac{88914}{143}a^{7}+\frac{19387}{13}a^{6}+\frac{150883}{143}a^{5}-\frac{36886}{11}a^{4}+\frac{3544}{11}a^{3}+\frac{396797}{143}a^{2}-\frac{267840}{143}a+\frac{51495}{143}$, $\frac{16}{11}a^{13}-\frac{51}{143}a^{12}-\frac{6555}{143}a^{11}+\frac{184}{143}a^{10}+\frac{6015}{13}a^{9}-\frac{6379}{143}a^{8}-\frac{288095}{143}a^{7}+\frac{9136}{13}a^{6}+\frac{561695}{143}a^{5}-\frac{408185}{143}a^{4}-\frac{339814}{143}a^{3}+\frac{478063}{143}a^{2}-\frac{182803}{143}a+\frac{23021}{143}$, $\frac{60}{143}a^{13}-\frac{53}{143}a^{12}-\frac{1941}{143}a^{11}+\frac{1236}{143}a^{10}+\frac{1895}{13}a^{9}-\frac{12853}{143}a^{8}-\frac{7550}{11}a^{7}+\frac{6550}{13}a^{6}+\frac{202588}{143}a^{5}-\frac{200061}{143}a^{4}-\frac{115919}{143}a^{3}+\frac{208017}{143}a^{2}-\frac{85907}{143}a+\frac{11241}{143}$, $\frac{240}{143}a^{13}-\frac{366}{143}a^{12}-\frac{7588}{143}a^{11}+\frac{9817}{143}a^{10}+\frac{7206}{13}a^{9}-\frac{102166}{143}a^{8}-\frac{355332}{143}a^{7}+\frac{46992}{13}a^{6}+\frac{642943}{143}a^{5}-\frac{1246690}{143}a^{4}-\frac{93064}{143}a^{3}+\frac{1099291}{143}a^{2}-\frac{665059}{143}a+\frac{121689}{143}$, $\frac{14}{13}a^{13}+\frac{3}{13}a^{12}-\frac{437}{13}a^{11}-\frac{188}{13}a^{10}+\frac{4282}{13}a^{9}+\frac{1503}{13}a^{8}-\frac{17926}{13}a^{7}-89a^{6}+\frac{34448}{13}a^{5}-942a^{4}-\frac{24217}{13}a^{3}+1581a^{2}-\frac{4974}{13}a+\frac{223}{13}$, $\frac{391}{143}a^{13}-\frac{563}{143}a^{12}-\frac{12447}{143}a^{11}+\frac{14901}{143}a^{10}+\frac{11952}{13}a^{9}-\frac{153951}{143}a^{8}-\frac{600504}{143}a^{7}+\frac{70852}{13}a^{6}+\frac{1136244}{143}a^{5}-\frac{1898519}{143}a^{4}-\frac{325115}{143}a^{3}+\frac{1714871}{143}a^{2}-\frac{946025}{143}a+\frac{160931}{143}$
| 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: | \( 26761483.3151 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{0}\cdot 26761483.3151 \cdot 1}{2\cdot\sqrt{153645020121345627783168}}\cr\approx \mathstrut & 0.559294927864 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 31*x^12 + 56*x^11 + 314*x^10 - 584*x^9 - 1306*x^8 + 2870*x^7 + 1747*x^6 - 6530*x^5 + 1987*x^4 + 4874*x^3 - 4973*x^2 + 1816*x - 238)
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 - 2*x^13 - 31*x^12 + 56*x^11 + 314*x^10 - 584*x^9 - 1306*x^8 + 2870*x^7 + 1747*x^6 - 6530*x^5 + 1987*x^4 + 4874*x^3 - 4973*x^2 + 1816*x - 238, 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 - 2*x^13 - 31*x^12 + 56*x^11 + 314*x^10 - 584*x^9 - 1306*x^8 + 2870*x^7 + 1747*x^6 - 6530*x^5 + 1987*x^4 + 4874*x^3 - 4973*x^2 + 1816*x - 238);
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 - 2*x^13 - 31*x^12 + 56*x^11 + 314*x^10 - 584*x^9 - 1306*x^8 + 2870*x^7 + 1747*x^6 - 6530*x^5 + 1987*x^4 + 4874*x^3 - 4973*x^2 + 1816*x - 238);
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
$C_2\times F_7$ (as 14T7):
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 84 |
| The 14 conjugacy class representatives for $F_7 \times C_2$ |
| Character table for $F_7 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 7.7.138584369664.1 |
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
| Degree 14 sibling: | data not computed |
| Degree 28 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 | R | R | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(3\)
| 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.12.10.1 | $x^{12} + 72 x^{11} + 2172 x^{10} + 35280 x^{9} + 328380 x^{8} + 1703232 x^{7} + 4282170 x^{6} + 3407400 x^{5} + 1340820 x^{4} + 712800 x^{3} + 3855192 x^{2} + 18082080 x + 35650393$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |