Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171)
gp: K = bnfinit(y^14 - 812*y^11 + 8729*y^10 + 239134*y^9 + 5865685*y^8 + 70091666*y^7 + 733767454*y^6 + 5616786700*y^5 + 39308800027*y^4 + 200176985575*y^3 + 941525394390*y^2 + 2923659885741*y + 7775628764171, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171)
\( x^{14} - 812 x^{11} + 8729 x^{10} + 239134 x^{9} + 5865685 x^{8} + 70091666 x^{7} + 733767454 x^{6} + \cdots + 7775628764171 \)
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: |
\(-474489902930063357496193840307474416087\)
\(\medspace = -\,7^{25}\cdot 29^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(578.88\) | 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: | $7^{25/14}29^{6/7}\approx 578.8787949105161$ | ||
| Ramified primes: |
\(7\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $14$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1421=7^{2}\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1421}(1184,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(923,·)$, $\chi_{1421}(36,·)$, $\chi_{1421}(993,·)$, $\chi_{1421}(1205,·)$, $\chi_{1421}(1415,·)$, $\chi_{1421}(750,·)$, $\chi_{1421}(1296,·)$, $\chi_{1421}(146,·)$, $\chi_{1421}(83,·)$, $\chi_{1421}(545,·)$, $\chi_{1421}(1147,·)$, $\chi_{1421}(223,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{64}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{29}a^{7}$, $\frac{1}{29}a^{8}$, $\frac{1}{29}a^{9}$, $\frac{1}{841}a^{10}+\frac{11}{29}a^{6}+\frac{10}{29}a^{5}-\frac{10}{29}a^{4}+\frac{7}{29}a^{3}$, $\frac{1}{841}a^{11}+\frac{10}{29}a^{6}-\frac{10}{29}a^{5}+\frac{7}{29}a^{4}$, $\frac{1}{24389}a^{12}+\frac{1}{841}a^{9}+\frac{11}{841}a^{8}+\frac{10}{841}a^{7}-\frac{416}{841}a^{6}-\frac{80}{841}a^{5}$, $\frac{1}{25\!\cdots\!61}a^{13}-\frac{39\!\cdots\!74}{25\!\cdots\!61}a^{12}+\frac{41\!\cdots\!46}{87\!\cdots\!09}a^{11}+\frac{10\!\cdots\!45}{87\!\cdots\!09}a^{10}-\frac{98\!\cdots\!22}{87\!\cdots\!09}a^{9}-\frac{72\!\cdots\!12}{87\!\cdots\!09}a^{8}+\frac{14\!\cdots\!41}{87\!\cdots\!09}a^{7}-\frac{33\!\cdots\!83}{87\!\cdots\!09}a^{6}-\frac{28\!\cdots\!81}{87\!\cdots\!09}a^{5}+\frac{51\!\cdots\!00}{30\!\cdots\!21}a^{4}-\frac{31\!\cdots\!89}{30\!\cdots\!21}a^{3}+\frac{41\!\cdots\!45}{10\!\cdots\!49}a^{2}-\frac{44\!\cdots\!40}{10\!\cdots\!49}a+\frac{17\!\cdots\!63}{10\!\cdots\!49}$
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
$C_{3}\times C_{3}\times C_{3}\times C_{21}\times C_{21}\times C_{21}$, which has order $250047$ (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: |
$\frac{35\!\cdots\!91}{39\!\cdots\!63}a^{13}+\frac{53\!\cdots\!25}{39\!\cdots\!63}a^{12}-\frac{63\!\cdots\!83}{13\!\cdots\!47}a^{11}-\frac{10\!\cdots\!92}{13\!\cdots\!47}a^{10}+\frac{11\!\cdots\!23}{13\!\cdots\!47}a^{9}+\frac{36\!\cdots\!00}{13\!\cdots\!47}a^{8}+\frac{71\!\cdots\!35}{13\!\cdots\!47}a^{7}+\frac{79\!\cdots\!55}{13\!\cdots\!47}a^{6}+\frac{66\!\cdots\!36}{13\!\cdots\!47}a^{5}+\frac{14\!\cdots\!86}{47\!\cdots\!43}a^{4}+\frac{76\!\cdots\!74}{47\!\cdots\!43}a^{3}+\frac{97\!\cdots\!57}{16\!\cdots\!67}a^{2}+\frac{24\!\cdots\!38}{16\!\cdots\!67}a+\frac{66\!\cdots\!61}{16\!\cdots\!67}$, $\frac{10\!\cdots\!57}{39\!\cdots\!63}a^{13}-\frac{47\!\cdots\!70}{39\!\cdots\!63}a^{12}-\frac{13\!\cdots\!79}{13\!\cdots\!47}a^{11}-\frac{30\!\cdots\!40}{13\!\cdots\!47}a^{10}+\frac{44\!\cdots\!41}{13\!\cdots\!47}a^{9}+\frac{83\!\cdots\!89}{13\!\cdots\!47}a^{8}+\frac{15\!\cdots\!81}{13\!\cdots\!47}a^{7}+\frac{14\!\cdots\!36}{13\!\cdots\!47}a^{6}+\frac{12\!\cdots\!43}{13\!\cdots\!47}a^{5}+\frac{26\!\cdots\!62}{47\!\cdots\!43}a^{4}+\frac{13\!\cdots\!65}{47\!\cdots\!43}a^{3}+\frac{17\!\cdots\!68}{16\!\cdots\!67}a^{2}+\frac{50\!\cdots\!59}{16\!\cdots\!67}a+\frac{83\!\cdots\!75}{16\!\cdots\!67}$, $\frac{74\!\cdots\!24}{39\!\cdots\!63}a^{13}-\frac{12\!\cdots\!54}{39\!\cdots\!63}a^{12}+\frac{16\!\cdots\!06}{47\!\cdots\!43}a^{11}-\frac{30\!\cdots\!01}{13\!\cdots\!47}a^{10}+\frac{37\!\cdots\!13}{13\!\cdots\!47}a^{9}-\frac{51\!\cdots\!12}{13\!\cdots\!47}a^{8}-\frac{47\!\cdots\!99}{13\!\cdots\!47}a^{7}-\frac{13\!\cdots\!73}{13\!\cdots\!47}a^{6}-\frac{84\!\cdots\!47}{13\!\cdots\!47}a^{5}-\frac{35\!\cdots\!61}{47\!\cdots\!43}a^{4}-\frac{16\!\cdots\!69}{47\!\cdots\!43}a^{3}-\frac{40\!\cdots\!16}{16\!\cdots\!67}a^{2}-\frac{11\!\cdots\!69}{16\!\cdots\!67}a-\frac{58\!\cdots\!06}{16\!\cdots\!67}$, $\frac{10\!\cdots\!89}{39\!\cdots\!63}a^{13}-\frac{43\!\cdots\!44}{39\!\cdots\!63}a^{12}+\frac{16\!\cdots\!35}{13\!\cdots\!47}a^{11}-\frac{12\!\cdots\!61}{13\!\cdots\!47}a^{10}+\frac{14\!\cdots\!03}{13\!\cdots\!47}a^{9}-\frac{11\!\cdots\!51}{13\!\cdots\!47}a^{8}+\frac{90\!\cdots\!29}{13\!\cdots\!47}a^{7}-\frac{25\!\cdots\!47}{13\!\cdots\!47}a^{6}-\frac{10\!\cdots\!43}{13\!\cdots\!47}a^{5}-\frac{80\!\cdots\!84}{47\!\cdots\!43}a^{4}-\frac{33\!\cdots\!02}{47\!\cdots\!43}a^{3}-\frac{10\!\cdots\!34}{16\!\cdots\!67}a^{2}-\frac{29\!\cdots\!18}{16\!\cdots\!67}a-\frac{19\!\cdots\!59}{16\!\cdots\!67}$, $\frac{11\!\cdots\!58}{39\!\cdots\!63}a^{13}-\frac{67\!\cdots\!10}{39\!\cdots\!63}a^{12}-\frac{83\!\cdots\!74}{13\!\cdots\!47}a^{11}-\frac{39\!\cdots\!10}{13\!\cdots\!47}a^{10}+\frac{42\!\cdots\!20}{13\!\cdots\!47}a^{9}+\frac{10\!\cdots\!17}{13\!\cdots\!47}a^{8}+\frac{21\!\cdots\!99}{13\!\cdots\!47}a^{7}+\frac{23\!\cdots\!10}{13\!\cdots\!47}a^{6}+\frac{20\!\cdots\!72}{13\!\cdots\!47}a^{5}+\frac{43\!\cdots\!45}{47\!\cdots\!43}a^{4}+\frac{25\!\cdots\!30}{47\!\cdots\!43}a^{3}+\frac{34\!\cdots\!45}{16\!\cdots\!67}a^{2}+\frac{10\!\cdots\!06}{16\!\cdots\!67}a+\frac{77\!\cdots\!90}{16\!\cdots\!67}$, $\frac{13\!\cdots\!27}{39\!\cdots\!63}a^{13}+\frac{35\!\cdots\!09}{39\!\cdots\!63}a^{12}-\frac{25\!\cdots\!93}{13\!\cdots\!47}a^{11}-\frac{30\!\cdots\!25}{13\!\cdots\!47}a^{10}+\frac{33\!\cdots\!13}{13\!\cdots\!47}a^{9}+\frac{14\!\cdots\!56}{13\!\cdots\!47}a^{8}+\frac{28\!\cdots\!36}{13\!\cdots\!47}a^{7}+\frac{35\!\cdots\!72}{13\!\cdots\!47}a^{6}+\frac{31\!\cdots\!32}{13\!\cdots\!47}a^{5}+\frac{80\!\cdots\!85}{47\!\cdots\!43}a^{4}+\frac{47\!\cdots\!62}{47\!\cdots\!43}a^{3}+\frac{80\!\cdots\!74}{16\!\cdots\!67}a^{2}+\frac{25\!\cdots\!69}{16\!\cdots\!67}a+\frac{89\!\cdots\!86}{16\!\cdots\!67}$
| 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: | \( 59811086.44175506 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 59811086.44175506 \cdot 250047}{2\cdot\sqrt{474489902930063357496193840307474416087}}\cr\approx \mathstrut & 0.132714671028730 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171)
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 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171, 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 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171);
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 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171);
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 cyclic group of order 14 |
| The 14 conjugacy class representatives for $C_{14}$ |
| Character table for $C_{14}$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), 7.7.8233120419813614521.6 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | R | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.14.0.1}{14} }$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.14.25.52 | $x^{14} + 7 x^{13} + 42 x^{12} + 42 x^{7} + 203$ | $14$ | $1$ | $25$ | $C_{14}$ | $[2]_{2}$ |
|
\(29\)
| 29.7.6.3 | $x^{7} + 87$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.7.6.3 | $x^{7} + 87$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |