Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22)
gp: K = bnfinit(y^14 - y^13 + y^12 + 7*y^11 + 9*y^10 - 5*y^9 + 21*y^8 + 63*y^7 + 32*y^6 + 74*y^5 + 188*y^4 + 70*y^3 - 230*y^2 - 144*y - 22, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22)
\( x^{14} - x^{13} + x^{12} + 7 x^{11} + 9 x^{10} - 5 x^{9} + 21 x^{8} + 63 x^{7} + 32 x^{6} + 74 x^{5} + \cdots - 22 \)
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: |
\(4156382630830772224\)
\(\medspace = 2^{12}\cdot 317^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.38\) | 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: | $2^{6/7}317^{1/2}\approx 32.251902756545775$ | ||
| Ramified primes: |
\(2\), \(317\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{9}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{16}a^{8}+\frac{5}{16}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{8}$, $\frac{1}{42045951392}a^{13}+\frac{280105071}{10511487848}a^{12}-\frac{1674539243}{42045951392}a^{11}+\frac{260012967}{2627871962}a^{10}-\frac{8463272311}{42045951392}a^{9}-\frac{1278847713}{2627871962}a^{8}-\frac{7847461091}{42045951392}a^{7}-\frac{136497702}{1313935981}a^{6}-\frac{87970726}{1313935981}a^{5}+\frac{5876552373}{21022975696}a^{4}+\frac{507119023}{21022975696}a^{3}+\frac{154263775}{10511487848}a^{2}+\frac{9430707683}{21022975696}a-\frac{5858881825}{21022975696}$
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
$C_{3}$, which has order $3$
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: |
$\frac{2752083745}{10511487848}a^{13}-\frac{7393569175}{21022975696}a^{12}+\frac{8148949719}{21022975696}a^{11}+\frac{17736143893}{10511487848}a^{10}+\frac{9451697525}{5255743924}a^{9}-\frac{40315100957}{21022975696}a^{8}+\frac{129461775845}{21022975696}a^{7}+\frac{37834793203}{2627871962}a^{6}+\frac{9142920445}{2627871962}a^{5}+\frac{95955329219}{5255743924}a^{4}+\frac{450880387439}{10511487848}a^{3}+\frac{10997789283}{2627871962}a^{2}-\frac{161535884489}{2627871962}a-\frac{186070006505}{10511487848}$, $\frac{3595714437}{42045951392}a^{13}-\frac{1199553961}{10511487848}a^{12}+\frac{5309226017}{42045951392}a^{11}+\frac{2935016745}{5255743924}a^{10}+\frac{24627767445}{42045951392}a^{9}-\frac{3221663471}{5255743924}a^{8}+\frac{86476635745}{42045951392}a^{7}+\frac{12672009969}{2627871962}a^{6}+\frac{2870142305}{2627871962}a^{5}+\frac{128007779137}{21022975696}a^{4}+\frac{307181690755}{21022975696}a^{3}+\frac{23144475851}{10511487848}a^{2}-\frac{423374889705}{21022975696}a-\frac{116940427317}{21022975696}$, $\frac{14095761013}{42045951392}a^{13}-\frac{9215908427}{21022975696}a^{12}+\frac{19472875091}{42045951392}a^{11}+\frac{23314273737}{10511487848}a^{10}+\frac{97260960265}{42045951392}a^{9}-\frac{51013117935}{21022975696}a^{8}+\frac{327350567391}{42045951392}a^{7}+\frac{49100513061}{2627871962}a^{6}+\frac{12793321743}{2627871962}a^{5}+\frac{480153597249}{21022975696}a^{4}+\frac{1179345180569}{21022975696}a^{3}+\frac{56910359389}{10511487848}a^{2}-\frac{1678476603977}{21022975696}a-\frac{504093113483}{21022975696}$, $\frac{4855273145}{42045951392}a^{13}-\frac{1562176289}{10511487848}a^{12}+\frac{6487974449}{42045951392}a^{11}+\frac{2050773155}{2627871962}a^{10}+\frac{33107685161}{42045951392}a^{9}-\frac{2053779341}{2627871962}a^{8}+\frac{112645864257}{42045951392}a^{7}+\frac{17312475703}{2627871962}a^{6}+\frac{4621668161}{2627871962}a^{5}+\frac{175332271589}{21022975696}a^{4}+\frac{414554009391}{21022975696}a^{3}+\frac{26908358229}{10511487848}a^{2}-\frac{558971975537}{21022975696}a-\frac{174861901845}{21022975696}$, $\frac{5641681497}{42045951392}a^{13}-\frac{3580481837}{21022975696}a^{12}+\frac{7762310087}{42045951392}a^{11}+\frac{9102437669}{10511487848}a^{10}+\frac{42954504293}{42045951392}a^{9}-\frac{20898878345}{21022975696}a^{8}+\frac{129661407339}{42045951392}a^{7}+\frac{19901816477}{2627871962}a^{6}+\frac{6256233863}{2627871962}a^{5}+\frac{191370007285}{21022975696}a^{4}+\frac{472446762177}{21022975696}a^{3}+\frac{44765176699}{10511487848}a^{2}-\frac{691781319457}{21022975696}a-\frac{208534339239}{21022975696}$, $\frac{2642061819}{42045951392}a^{13}-\frac{1682568439}{21022975696}a^{12}+\frac{3412798025}{42045951392}a^{11}+\frac{4469535655}{10511487848}a^{10}+\frac{18484154471}{42045951392}a^{9}-\frac{10314340187}{21022975696}a^{8}+\frac{62880316029}{42045951392}a^{7}+\frac{4610679277}{1313935981}a^{6}+\frac{1031199039}{1313935981}a^{5}+\frac{91165740519}{21022975696}a^{4}+\frac{218064354603}{21022975696}a^{3}+\frac{5609135567}{10511487848}a^{2}-\frac{320352891679}{21022975696}a-\frac{97531250961}{21022975696}$, $\frac{4587258221}{42045951392}a^{13}-\frac{1565397671}{10511487848}a^{12}+\frac{7224672681}{42045951392}a^{11}+\frac{896542364}{1313935981}a^{10}+\frac{31922013349}{42045951392}a^{9}-\frac{4133245769}{5255743924}a^{8}+\frac{108340126401}{42045951392}a^{7}+\frac{7689728308}{1313935981}a^{6}+\frac{2135647802}{1313935981}a^{5}+\frac{156229968945}{21022975696}a^{4}+\frac{378088162779}{21022975696}a^{3}+\frac{17917020363}{10511487848}a^{2}-\frac{526186751481}{21022975696}a-\frac{159721709381}{21022975696}$
| 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: | \( 5502.73317638 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 5502.73317638 \cdot 3}{2\cdot\sqrt{4156382630830772224}}\cr\approx \mathstrut & 0.996440003506 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22)
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 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22, 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 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22);
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 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22);
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
$\GL(3,2)$ (as 14T10):
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 non-solvable group of order 168 |
| The 6 conjugacy class representatives for $\PSL(2,7)$ |
| Character table for $\PSL(2,7)$ |
Intermediate fields
| 7.3.6431296.2 |
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 7 siblings: | 7.3.6431296.2, 7.3.6431296.1 |
| Degree 8 sibling: | 8.0.646274503744.1 |
| Degree 21 sibling: | 21.5.26730926988131422081122304.1 |
| Degree 24 sibling: | data not computed |
| Degree 28 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Arithmetically equvalently sibling: | 14.2.4156382630830772224.1 |
| Minimal sibling: | 7.3.6431296.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
|
\(317\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |