Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 14*x^12 + 686*x^8 - 1540*x^6 - 13482*x^4 + 68068*x^2 - 110894)
gp: K = bnfinit(y^14 - 14*y^12 + 686*y^8 - 1540*y^6 - 13482*y^4 + 68068*y^2 - 110894, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 14*x^12 + 686*x^8 - 1540*x^6 - 13482*x^4 + 68068*x^2 - 110894);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 14*x^12 + 686*x^8 - 1540*x^6 - 13482*x^4 + 68068*x^2 - 110894)
\( x^{14} - 14x^{12} + 686x^{8} - 1540x^{6} - 13482x^{4} + 68068x^{2} - 110894 \)
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: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(338637882455405716008075264\)
\(\medspace = 2^{27}\cdot 3^{12}\cdot 7^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(78.52\) | 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: | $2^{27/14}3^{6/7}7^{47/42}\approx 86.14376989441227$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{14}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{33}a^{10}-\frac{4}{33}a^{8}+\frac{1}{33}a^{6}+\frac{4}{33}a^{4}-\frac{7}{33}a^{2}+\frac{10}{33}$, $\frac{1}{33}a^{11}-\frac{4}{33}a^{9}+\frac{1}{33}a^{7}+\frac{4}{33}a^{5}-\frac{7}{33}a^{3}+\frac{10}{33}a$, $\frac{1}{890294889}a^{12}-\frac{13448917}{890294889}a^{10}-\frac{19756078}{890294889}a^{8}+\frac{47364701}{890294889}a^{6}+\frac{349585253}{890294889}a^{4}-\frac{355303288}{890294889}a^{2}+\frac{58266400}{890294889}$, $\frac{1}{79236245121}a^{13}-\frac{1146551503}{79236245121}a^{11}-\frac{19319213}{890294889}a^{9}+\frac{25623108785}{79236245121}a^{7}+\frac{21635726690}{79236245121}a^{5}-\frac{31596560302}{79236245121}a^{3}+\frac{27009920767}{79236245121}a$
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: | $7$ | 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: |
$\frac{42386484}{26412081707}a^{13}+\frac{1851214}{890294889}a^{12}-\frac{1711732766}{79236245121}a^{11}-\frac{23629316}{890294889}a^{10}-\frac{12565478}{890294889}a^{9}-\frac{2556166}{296764963}a^{8}+\frac{92377459480}{79236245121}a^{7}+\frac{1208688886}{890294889}a^{6}-\frac{151449912932}{79236245121}a^{5}-\frac{2988506366}{890294889}a^{4}-\frac{2199717941518}{79236245121}a^{3}-\frac{9587441306}{296764963}a^{2}+\frac{7593210780154}{79236245121}a+\frac{44085102101}{296764963}$, $\frac{2942806}{79236245121}a^{13}-\frac{315754}{890294889}a^{12}-\frac{33923267}{79236245121}a^{11}+\frac{4589686}{890294889}a^{10}-\frac{2102219}{890294889}a^{9}+\frac{7173286}{890294889}a^{8}+\frac{204142144}{26412081707}a^{7}-\frac{258382334}{890294889}a^{6}+\frac{21006625111}{79236245121}a^{5}+\frac{27029206}{890294889}a^{4}-\frac{23740155760}{79236245121}a^{3}+\frac{4803513232}{890294889}a^{2}-\frac{447801679951}{79236245121}a-\frac{1847445991}{890294889}$, $\frac{2942806}{79236245121}a^{13}+\frac{315754}{890294889}a^{12}-\frac{33923267}{79236245121}a^{11}-\frac{4589686}{890294889}a^{10}-\frac{2102219}{890294889}a^{9}-\frac{7173286}{890294889}a^{8}+\frac{204142144}{26412081707}a^{7}+\frac{258382334}{890294889}a^{6}+\frac{21006625111}{79236245121}a^{5}-\frac{27029206}{890294889}a^{4}-\frac{23740155760}{79236245121}a^{3}-\frac{4803513232}{890294889}a^{2}-\frac{447801679951}{79236245121}a+\frac{1847445991}{890294889}$, $\frac{212972}{890294889}a^{12}-\frac{2221613}{890294889}a^{10}-\frac{3918556}{296764963}a^{8}+\frac{59705279}{296764963}a^{6}-\frac{172693295}{890294889}a^{4}-\frac{749406236}{296764963}a^{2}+\frac{3284120713}{890294889}$, $\frac{182484359}{79236245121}a^{13}+\frac{13330652}{890294889}a^{12}-\frac{25796008}{79236245121}a^{11}-\frac{130689637}{890294889}a^{10}-\frac{33743432}{296764963}a^{9}-\frac{217553455}{296764963}a^{8}-\frac{2401518814}{79236245121}a^{7}+\frac{6559767902}{890294889}a^{6}+\frac{128364945149}{79236245121}a^{5}+\frac{12239909063}{890294889}a^{4}-\frac{120823068149}{26412081707}a^{3}-\frac{42776920815}{296764963}a^{2}-\frac{13599086494}{26412081707}a+\frac{99240706577}{296764963}$, $\frac{5626282}{890294889}a^{12}-\frac{16653544}{296764963}a^{10}-\frac{245915309}{890294889}a^{8}+\frac{871557129}{296764963}a^{6}+\frac{1262344428}{296764963}a^{4}-\frac{59456898257}{890294889}a^{2}+\frac{102384469307}{890294889}$, $\frac{191927245919}{79236245121}a^{13}-\frac{5584564817}{890294889}a^{12}-\frac{1317273246362}{79236245121}a^{11}+\frac{12142992883}{296764963}a^{10}-\frac{36119536285}{296764963}a^{9}+\frac{311923111606}{890294889}a^{8}+\frac{21004475052652}{26412081707}a^{7}-\frac{666521903822}{296764963}a^{6}+\frac{202873617701668}{79236245121}a^{5}-\frac{1884392241196}{296764963}a^{4}-\frac{522695668725442}{26412081707}a^{3}+\frac{45970835860492}{890294889}a^{2}+\frac{32\!\cdots\!95}{79236245121}a-\frac{96018577115287}{890294889}$
| 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: | \( 129960074.14382683 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 129960074.14382683 \cdot 1}{2\cdot\sqrt{338637882455405716008075264}}\cr\approx \mathstrut & 0.869063252056408 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 14*x^12 + 686*x^8 - 1540*x^6 - 13482*x^4 + 68068*x^2 - 110894)
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 - 14*x^12 + 686*x^8 - 1540*x^6 - 13482*x^4 + 68068*x^2 - 110894, 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 - 14*x^12 + 686*x^8 - 1540*x^6 - 13482*x^4 + 68068*x^2 - 110894);
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 - 14*x^12 + 686*x^8 - 1540*x^6 - 13482*x^4 + 68068*x^2 - 110894);
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
| \(\Q(\sqrt{14}) \), 7.1.38423222208.1 |
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: | 14.0.48376840350772245144010752.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{7}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.14.27.121 | $x^{14} + 6 x^{12} + 4 x^{9} + 4 x^{7} + 4 x^{4} + 6$ | $14$ | $1$ | $27$ | $(C_7:C_3) \times C_2$ | $[3]_{7}^{3}$ |
|
\(3\)
| 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
|
\(7\)
| 7.14.15.1 | $x^{14} + 14 x^{3} + 7 x^{2} + 21$ | $14$ | $1$ | $15$ | $F_7 \times C_2$ | $[7/6]_{6}^{2}$ |