Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 22*x^12 - 3*x^11 + 162*x^10 + 12*x^9 - 485*x^8 + 81*x^7 + 591*x^6 - 246*x^5 - 219*x^4 + 145*x^3 - 10*x^2 - 7*x + 1)
gp: K = bnfinit(y^14 - 22*y^12 - 3*y^11 + 162*y^10 + 12*y^9 - 485*y^8 + 81*y^7 + 591*y^6 - 246*y^5 - 219*y^4 + 145*y^3 - 10*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 22*x^12 - 3*x^11 + 162*x^10 + 12*x^9 - 485*x^8 + 81*x^7 + 591*x^6 - 246*x^5 - 219*x^4 + 145*x^3 - 10*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 22*x^12 - 3*x^11 + 162*x^10 + 12*x^9 - 485*x^8 + 81*x^7 + 591*x^6 - 246*x^5 - 219*x^4 + 145*x^3 - 10*x^2 - 7*x + 1)
\( x^{14} - 22 x^{12} - 3 x^{11} + 162 x^{10} + 12 x^{9} - 485 x^{8} + 81 x^{7} + 591 x^{6} - 246 x^{5} + \cdots + 1 \)
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: |
\(26736653147041339876997\)
\(\medspace = 7^{8}\cdot 173^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.99\) | 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: | $7^{2/3}173^{1/2}\approx 48.13065200407494$ | ||
| Ramified primes: |
\(7\), \(173\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{173}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3196229}a^{13}-\frac{260888}{3196229}a^{12}-\frac{1148033}{3196229}a^{11}-\frac{997602}{3196229}a^{10}-\frac{144274}{3196229}a^{9}+\frac{562620}{3196229}a^{8}-\frac{382678}{3196229}a^{7}-\frac{1310899}{3196229}a^{6}+\frac{1315903}{3196229}a^{5}+\frac{458551}{3196229}a^{4}+\frac{1201734}{3196229}a^{3}+\frac{122963}{3196229}a^{2}+\frac{979319}{3196229}a+\frac{1186065}{3196229}$
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: |
$\frac{685368}{3196229}a^{13}-\frac{844066}{3196229}a^{12}-\frac{15780672}{3196229}a^{11}+\frac{16016373}{3196229}a^{10}+\frac{128802901}{3196229}a^{9}-\frac{115975331}{3196229}a^{8}-\frac{449764511}{3196229}a^{7}+\frac{385704861}{3196229}a^{6}+\frac{633919945}{3196229}a^{5}-\frac{536393587}{3196229}a^{4}-\frac{290687279}{3196229}a^{3}+\frac{239652516}{3196229}a^{2}+\frac{11384224}{3196229}a-\frac{9120879}{3196229}$, $a$, $\frac{13295064}{3196229}a^{13}+\frac{5876594}{3196229}a^{12}-\frac{290182450}{3196229}a^{11}-\frac{168110250}{3196229}a^{10}+\frac{2085610168}{3196229}a^{9}+\frac{1081625191}{3196229}a^{8}-\frac{6012052002}{3196229}a^{7}-\frac{1581250279}{3196229}a^{6}+\frac{7266335291}{3196229}a^{5}-\frac{76638229}{3196229}a^{4}-\frac{3036512950}{3196229}a^{3}+\frac{607421680}{3196229}a^{2}+\frac{158081901}{3196229}a-\frac{28601744}{3196229}$, $\frac{18142308}{3196229}a^{13}+\frac{6087772}{3196229}a^{12}-\frac{397228609}{3196229}a^{11}-\frac{187995206}{3196229}a^{10}+\frac{2878703075}{3196229}a^{9}+\frac{1189891839}{3196229}a^{8}-\frac{8414559263}{3196229}a^{7}-\frac{1392902819}{3196229}a^{6}+\frac{10276840155}{3196229}a^{5}-\frac{943019108}{3196229}a^{4}-\frac{4306081206}{3196229}a^{3}+\frac{1152856891}{3196229}a^{2}+\frac{216001036}{3196229}a-\frac{54313428}{3196229}$, $\frac{17282036}{3196229}a^{13}+\frac{7921386}{3196229}a^{12}-\frac{376186114}{3196229}a^{11}-\frac{223946084}{3196229}a^{10}+\frac{2689043564}{3196229}a^{9}+\frac{1431530073}{3196229}a^{8}-\frac{7674256490}{3196229}a^{7}-\frac{2063080680}{3196229}a^{6}+\frac{9148060464}{3196229}a^{5}-\frac{156749466}{3196229}a^{4}-\frac{3741386500}{3196229}a^{3}+\frac{834003039}{3196229}a^{2}+\frac{168560027}{3196229}a-\frac{38265835}{3196229}$, $a-1$, $\frac{13840055}{3196229}a^{13}+\frac{5315422}{3196229}a^{12}-\frac{302400861}{3196229}a^{11}-\frac{157687787}{3196229}a^{10}+\frac{2180698904}{3196229}a^{9}+\frac{1003915458}{3196229}a^{8}-\frac{6321013634}{3196229}a^{7}-\frac{1307032956}{3196229}a^{6}+\frac{7660881456}{3196229}a^{5}-\frac{467163178}{3196229}a^{4}-\frac{3188609480}{3196229}a^{3}+\frac{791197852}{3196229}a^{2}+\frac{152627236}{3196229}a-\frac{39204289}{3196229}$, $\frac{6208757}{3196229}a^{13}+\frac{1128862}{3196229}a^{12}-\frac{136209592}{3196229}a^{11}-\frac{43248148}{3196229}a^{10}+\frac{994174425}{3196229}a^{9}+\frac{251906415}{3196229}a^{8}-\frac{2940060028}{3196229}a^{7}-\frac{12809264}{3196229}a^{6}+\frac{3603448579}{3196229}a^{5}-\frac{891409847}{3196229}a^{4}-\frac{1460073386}{3196229}a^{3}+\frac{622588935}{3196229}a^{2}+\frac{36224394}{3196229}a-\frac{29554154}{3196229}$, $\frac{514331}{3196229}a^{13}+\frac{1299950}{3196229}a^{12}-\frac{10400379}{3196229}a^{11}-\frac{29366495}{3196229}a^{10}+\frac{59790121}{3196229}a^{9}+\frac{197284674}{3196229}a^{8}-\frac{98459697}{3196229}a^{7}-\frac{461331682}{3196229}a^{6}+\frac{46569891}{3196229}a^{5}+\frac{403177554}{3196229}a^{4}-\frac{20087669}{3196229}a^{3}-\frac{102379798}{3196229}a^{2}+\frac{9981166}{3196229}a+\frac{4123033}{3196229}$, $\frac{10558948}{3196229}a^{13}+\frac{5492574}{3196229}a^{12}-\frac{229576143}{3196229}a^{11}-\frac{151331441}{3196229}a^{10}+\frac{1635079018}{3196229}a^{9}+\frac{982338984}{3196229}a^{8}-\frac{4635736765}{3196229}a^{7}-\frac{1588939734}{3196229}a^{6}+\frac{5498041994}{3196229}a^{5}+\frac{331482827}{3196229}a^{4}-\frac{2261225445}{3196229}a^{3}+\frac{313793902}{3196229}a^{2}+\frac{115962009}{3196229}a-\frac{9638485}{3196229}$, $\frac{24865396}{3196229}a^{13}+\frac{8516584}{3196229}a^{12}-\frac{543838580}{3196229}a^{11}-\frac{260609849}{3196229}a^{10}+\frac{3932667621}{3196229}a^{9}+\frac{1639082928}{3196229}a^{8}-\frac{11453078988}{3196229}a^{7}-\frac{1867043765}{3196229}a^{6}+\frac{13926858625}{3196229}a^{5}-\frac{1431251401}{3196229}a^{4}-\frac{5786242261}{3196229}a^{3}+\frac{1673066028}{3196229}a^{2}+\frac{268599054}{3196229}a-\frac{82940778}{3196229}$, $\frac{2014199}{3196229}a^{13}+\frac{2072491}{3196229}a^{12}-\frac{42265601}{3196229}a^{11}-\frac{50001261}{3196229}a^{10}+\frac{276273619}{3196229}a^{9}+\frac{318879872}{3196229}a^{8}-\frac{651874914}{3196229}a^{7}-\frac{578802992}{3196229}a^{6}+\frac{570556064}{3196229}a^{5}+\frac{267154755}{3196229}a^{4}-\frac{110238481}{3196229}a^{3}+\frac{18540030}{3196229}a^{2}-\frac{12573098}{3196229}a+\frac{710549}{3196229}$, $\frac{17865752}{3196229}a^{13}+\frac{7752283}{3196229}a^{12}-\frac{389633289}{3196229}a^{11}-\frac{222429835}{3196229}a^{10}+\frac{2796914858}{3196229}a^{9}+\frac{1422903785}{3196229}a^{8}-\frac{8045434379}{3196229}a^{7}-\frac{2011457955}{3196229}a^{6}+\frac{9696890349}{3196229}a^{5}-\frac{254082196}{3196229}a^{4}-\frac{4046847112}{3196229}a^{3}+\frac{879702329}{3196229}a^{2}+\frac{211919987}{3196229}a-\frac{45115756}{3196229}$
| 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: | \( 5466417.1943 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{0}\cdot 5466417.1943 \cdot 1}{2\cdot\sqrt{26736653147041339876997}}\cr\approx \mathstrut & 0.27386667961 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 22*x^12 - 3*x^11 + 162*x^10 + 12*x^9 - 485*x^8 + 81*x^7 + 591*x^6 - 246*x^5 - 219*x^4 + 145*x^3 - 10*x^2 - 7*x + 1)
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 - 22*x^12 - 3*x^11 + 162*x^10 + 12*x^9 - 485*x^8 + 81*x^7 + 591*x^6 - 246*x^5 - 219*x^4 + 145*x^3 - 10*x^2 - 7*x + 1, 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 - 22*x^12 - 3*x^11 + 162*x^10 + 12*x^9 - 485*x^8 + 81*x^7 + 591*x^6 - 246*x^5 - 219*x^4 + 145*x^3 - 10*x^2 - 7*x + 1);
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 - 22*x^12 - 3*x^11 + 162*x^10 + 12*x^9 - 485*x^8 + 81*x^7 + 591*x^6 - 246*x^5 - 219*x^4 + 145*x^3 - 10*x^2 - 7*x + 1);
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
| \(\Q(\sqrt{173}) \), 7.7.12431698517.1 |
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
| Galois closure: | deg 42 |
| Degree 7 sibling: | 7.7.12431698517.1 |
| Degree 21 sibling: | 21.21.94142881806955162927406195366237.1 |
| Minimal sibling: | 7.7.12431698517.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(173\)
| 173.2.1.1 | $x^{2} + 173$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 173.6.3.1 | $x^{6} + 83040 x^{5} + 2298547723 x^{4} + 21207957785622 x^{3} + 402260049631 x^{2} + 42811394473266 x + 3626818487364574$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 173.6.3.1 | $x^{6} + 83040 x^{5} + 2298547723 x^{4} + 21207957785622 x^{3} + 402260049631 x^{2} + 42811394473266 x + 3626818487364574$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |