Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 18*x^12 - 75*x^11 + 243*x^10 - 1050*x^9 - 4659*x^8 + 19779*x^7 - 27675*x^6 + 175446*x^5 + 31257*x^4 - 1629999*x^3 + 2490912*x^2 - 1485693*x + 54279)
gp: K = bnfinit(y^14 - 3*y^13 + 18*y^12 - 75*y^11 + 243*y^10 - 1050*y^9 - 4659*y^8 + 19779*y^7 - 27675*y^6 + 175446*y^5 + 31257*y^4 - 1629999*y^3 + 2490912*y^2 - 1485693*y + 54279, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 + 18*x^12 - 75*x^11 + 243*x^10 - 1050*x^9 - 4659*x^8 + 19779*x^7 - 27675*x^6 + 175446*x^5 + 31257*x^4 - 1629999*x^3 + 2490912*x^2 - 1485693*x + 54279);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 + 18*x^12 - 75*x^11 + 243*x^10 - 1050*x^9 - 4659*x^8 + 19779*x^7 - 27675*x^6 + 175446*x^5 + 31257*x^4 - 1629999*x^3 + 2490912*x^2 - 1485693*x + 54279)
\( x^{14} - 3 x^{13} + 18 x^{12} - 75 x^{11} + 243 x^{10} - 1050 x^{9} - 4659 x^{8} + 19779 x^{7} + \cdots + 54279 \)
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: |
\(83876874860775669925367808\)
\(\medspace = 2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 11^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(71.07\) | 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: | $2^{6/7}3^{6/7}7^{5/6}11^{1/2}\approx 77.97071114609403$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{77}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{42\!\cdots\!71}a^{13}-\frac{11\!\cdots\!88}{14\!\cdots\!57}a^{12}+\frac{21\!\cdots\!40}{14\!\cdots\!57}a^{11}-\frac{18\!\cdots\!37}{14\!\cdots\!57}a^{10}+\frac{42\!\cdots\!21}{14\!\cdots\!57}a^{9}+\frac{11\!\cdots\!74}{14\!\cdots\!57}a^{8}-\frac{18\!\cdots\!06}{14\!\cdots\!57}a^{7}-\frac{13\!\cdots\!06}{14\!\cdots\!57}a^{6}+\frac{24\!\cdots\!82}{47\!\cdots\!19}a^{5}-\frac{19\!\cdots\!61}{47\!\cdots\!19}a^{4}-\frac{14\!\cdots\!18}{47\!\cdots\!19}a^{3}-\frac{10\!\cdots\!29}{47\!\cdots\!19}a^{2}-\frac{21\!\cdots\!86}{47\!\cdots\!19}a-\frac{19\!\cdots\!77}{47\!\cdots\!19}$
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
$C_{2}$, which has order $2$ (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: |
$\frac{73\!\cdots\!70}{14\!\cdots\!57}a^{13}-\frac{10\!\cdots\!82}{14\!\cdots\!57}a^{12}+\frac{11\!\cdots\!41}{14\!\cdots\!57}a^{11}-\frac{12\!\cdots\!94}{47\!\cdots\!19}a^{10}+\frac{12\!\cdots\!67}{14\!\cdots\!57}a^{9}-\frac{20\!\cdots\!63}{47\!\cdots\!19}a^{8}-\frac{14\!\cdots\!91}{47\!\cdots\!19}a^{7}+\frac{23\!\cdots\!86}{47\!\cdots\!19}a^{6}-\frac{63\!\cdots\!12}{47\!\cdots\!19}a^{5}+\frac{38\!\cdots\!57}{47\!\cdots\!19}a^{4}+\frac{66\!\cdots\!32}{47\!\cdots\!19}a^{3}-\frac{29\!\cdots\!76}{47\!\cdots\!19}a^{2}-\frac{27\!\cdots\!32}{47\!\cdots\!19}a+\frac{28\!\cdots\!02}{47\!\cdots\!19}$, $\frac{77\!\cdots\!10}{14\!\cdots\!57}a^{13}-\frac{79\!\cdots\!53}{47\!\cdots\!19}a^{12}+\frac{12\!\cdots\!24}{14\!\cdots\!57}a^{11}-\frac{20\!\cdots\!98}{47\!\cdots\!19}a^{10}+\frac{17\!\cdots\!56}{14\!\cdots\!57}a^{9}-\frac{85\!\cdots\!41}{14\!\cdots\!57}a^{8}-\frac{12\!\cdots\!02}{47\!\cdots\!19}a^{7}+\frac{51\!\cdots\!12}{47\!\cdots\!19}a^{6}-\frac{41\!\cdots\!83}{47\!\cdots\!19}a^{5}+\frac{46\!\cdots\!05}{47\!\cdots\!19}a^{4}+\frac{12\!\cdots\!97}{47\!\cdots\!19}a^{3}-\frac{44\!\cdots\!26}{47\!\cdots\!19}a^{2}+\frac{43\!\cdots\!45}{47\!\cdots\!19}a+\frac{42\!\cdots\!41}{47\!\cdots\!19}$, $\frac{46\!\cdots\!69}{42\!\cdots\!71}a^{13}-\frac{46\!\cdots\!66}{14\!\cdots\!57}a^{12}+\frac{27\!\cdots\!67}{14\!\cdots\!57}a^{11}-\frac{38\!\cdots\!89}{47\!\cdots\!19}a^{10}+\frac{37\!\cdots\!95}{14\!\cdots\!57}a^{9}-\frac{16\!\cdots\!12}{14\!\cdots\!57}a^{8}-\frac{23\!\cdots\!93}{47\!\cdots\!19}a^{7}+\frac{30\!\cdots\!92}{14\!\cdots\!57}a^{6}-\frac{13\!\cdots\!17}{47\!\cdots\!19}a^{5}+\frac{90\!\cdots\!68}{47\!\cdots\!19}a^{4}+\frac{13\!\cdots\!74}{47\!\cdots\!19}a^{3}-\frac{83\!\cdots\!73}{47\!\cdots\!19}a^{2}+\frac{12\!\cdots\!63}{47\!\cdots\!19}a-\frac{71\!\cdots\!17}{47\!\cdots\!19}$, $\frac{24\!\cdots\!18}{37\!\cdots\!17}a^{13}-\frac{23\!\cdots\!56}{12\!\cdots\!39}a^{12}+\frac{48\!\cdots\!66}{41\!\cdots\!13}a^{11}-\frac{59\!\cdots\!12}{12\!\cdots\!39}a^{10}+\frac{19\!\cdots\!24}{12\!\cdots\!39}a^{9}-\frac{83\!\cdots\!25}{12\!\cdots\!39}a^{8}-\frac{40\!\cdots\!42}{12\!\cdots\!39}a^{7}+\frac{16\!\cdots\!54}{12\!\cdots\!39}a^{6}-\frac{66\!\cdots\!78}{41\!\cdots\!13}a^{5}+\frac{45\!\cdots\!50}{41\!\cdots\!13}a^{4}+\frac{15\!\cdots\!50}{41\!\cdots\!13}a^{3}-\frac{45\!\cdots\!20}{41\!\cdots\!13}a^{2}+\frac{67\!\cdots\!82}{41\!\cdots\!13}a-\frac{21\!\cdots\!31}{41\!\cdots\!13}$, $\frac{16\!\cdots\!13}{14\!\cdots\!57}a^{13}+\frac{44\!\cdots\!78}{14\!\cdots\!57}a^{12}-\frac{21\!\cdots\!17}{14\!\cdots\!57}a^{11}-\frac{21\!\cdots\!36}{47\!\cdots\!19}a^{10}-\frac{74\!\cdots\!47}{14\!\cdots\!57}a^{9}-\frac{15\!\cdots\!08}{14\!\cdots\!57}a^{8}-\frac{68\!\cdots\!53}{47\!\cdots\!19}a^{7}-\frac{11\!\cdots\!68}{47\!\cdots\!19}a^{6}+\frac{11\!\cdots\!24}{47\!\cdots\!19}a^{5}+\frac{14\!\cdots\!71}{47\!\cdots\!19}a^{4}+\frac{93\!\cdots\!05}{47\!\cdots\!19}a^{3}+\frac{65\!\cdots\!67}{47\!\cdots\!19}a^{2}-\frac{82\!\cdots\!63}{47\!\cdots\!19}a+\frac{37\!\cdots\!55}{47\!\cdots\!19}$, $\frac{58\!\cdots\!95}{42\!\cdots\!71}a^{13}-\frac{58\!\cdots\!80}{14\!\cdots\!57}a^{12}+\frac{11\!\cdots\!96}{47\!\cdots\!19}a^{11}-\frac{14\!\cdots\!81}{14\!\cdots\!57}a^{10}+\frac{47\!\cdots\!45}{14\!\cdots\!57}a^{9}-\frac{20\!\cdots\!46}{14\!\cdots\!57}a^{8}-\frac{90\!\cdots\!57}{14\!\cdots\!57}a^{7}+\frac{38\!\cdots\!13}{14\!\cdots\!57}a^{6}-\frac{17\!\cdots\!78}{47\!\cdots\!19}a^{5}+\frac{11\!\cdots\!74}{47\!\cdots\!19}a^{4}+\frac{20\!\cdots\!25}{47\!\cdots\!19}a^{3}-\frac{10\!\cdots\!18}{47\!\cdots\!19}a^{2}+\frac{16\!\cdots\!03}{47\!\cdots\!19}a-\frac{96\!\cdots\!28}{47\!\cdots\!19}$, $\frac{26\!\cdots\!72}{42\!\cdots\!71}a^{13}-\frac{22\!\cdots\!46}{14\!\cdots\!57}a^{12}+\frac{50\!\cdots\!78}{47\!\cdots\!19}a^{11}-\frac{60\!\cdots\!21}{14\!\cdots\!57}a^{10}+\frac{63\!\cdots\!44}{47\!\cdots\!19}a^{9}-\frac{85\!\cdots\!89}{14\!\cdots\!57}a^{8}-\frac{14\!\cdots\!21}{47\!\cdots\!19}a^{7}+\frac{15\!\cdots\!30}{14\!\cdots\!57}a^{6}-\frac{61\!\cdots\!87}{47\!\cdots\!19}a^{5}+\frac{49\!\cdots\!20}{47\!\cdots\!19}a^{4}+\frac{28\!\cdots\!80}{47\!\cdots\!19}a^{3}-\frac{45\!\cdots\!90}{47\!\cdots\!19}a^{2}+\frac{54\!\cdots\!00}{47\!\cdots\!19}a-\frac{29\!\cdots\!25}{47\!\cdots\!19}$
| 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: | \( 20851980.295041185 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 20851980.295041185 \cdot 2}{2\cdot\sqrt{83876874860775669925367808}}\cr\approx \mathstrut & 0.560357755281746 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 18*x^12 - 75*x^11 + 243*x^10 - 1050*x^9 - 4659*x^8 + 19779*x^7 - 27675*x^6 + 175446*x^5 + 31257*x^4 - 1629999*x^3 + 2490912*x^2 - 1485693*x + 54279)
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 - 3*x^13 + 18*x^12 - 75*x^11 + 243*x^10 - 1050*x^9 - 4659*x^8 + 19779*x^7 - 27675*x^6 + 175446*x^5 + 31257*x^4 - 1629999*x^3 + 2490912*x^2 - 1485693*x + 54279, 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 - 3*x^13 + 18*x^12 - 75*x^11 + 243*x^10 - 1050*x^9 - 4659*x^8 + 19779*x^7 - 27675*x^6 + 175446*x^5 + 31257*x^4 - 1629999*x^3 + 2490912*x^2 - 1485693*x + 54279);
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 - 3*x^13 + 18*x^12 - 75*x^11 + 243*x^10 - 1050*x^9 - 4659*x^8 + 19779*x^7 - 27675*x^6 + 175446*x^5 + 31257*x^4 - 1629999*x^3 + 2490912*x^2 - 1485693*x + 54279);
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
| \(\Q(\sqrt{77}) \), 7.1.784147392.1 |
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: | 14.0.11982410694396524275052544.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | R | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.14.12.1 | $x^{14} + 7 x^{13} + 28 x^{12} + 77 x^{11} + 161 x^{10} + 266 x^{9} + 357 x^{8} + 397 x^{7} + 371 x^{6} + 224 x^{5} + 21 x^{4} + 7 x^{3} + 70 x^{2} + 35 x + 7$ | $7$ | $2$ | $12$ | $(C_7:C_3) \times C_2$ | $[\ ]_{7}^{6}$ |
|
\(3\)
| 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
|
\(7\)
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
|
\(11\)
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |