Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 6*x^12 - 85*x^10 + 355*x^8 - 695*x^6 + 1091*x^4 - 609*x^2 + 535)
gp: K = bnfinit(y^14 + 6*y^12 - 85*y^10 + 355*y^8 - 695*y^6 + 1091*y^4 - 609*y^2 + 535, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 + 6*x^12 - 85*x^10 + 355*x^8 - 695*x^6 + 1091*x^4 - 609*x^2 + 535);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 + 6*x^12 - 85*x^10 + 355*x^8 - 695*x^6 + 1091*x^4 - 609*x^2 + 535)
\( x^{14} + 6x^{12} - 85x^{10} + 355x^{8} - 695x^{6} + 1091x^{4} - 609x^{2} + 535 \)
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: |
\(-12545167784987734375\)
\(\medspace = -\,5^{7}\cdot 107^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.13\) | 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: | $5^{1/2}107^{1/2}\approx 23.130067012440755$ | ||
| Ramified primes: |
\(5\), \(107\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-535}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $14$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15}a^{6}+\frac{7}{15}a^{4}+\frac{1}{15}a^{2}-\frac{1}{3}$, $\frac{1}{30}a^{7}+\frac{7}{30}a^{5}-\frac{1}{2}a^{4}+\frac{1}{30}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{30}a^{8}-\frac{1}{30}a^{6}-\frac{1}{2}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{13}{30}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{30}a^{9}-\frac{1}{30}a^{6}+\frac{2}{5}a^{5}+\frac{4}{15}a^{4}-\frac{2}{5}a^{3}+\frac{7}{15}a^{2}+\frac{1}{6}a+\frac{1}{6}$, $\frac{1}{150}a^{10}-\frac{1}{150}a^{8}+\frac{1}{150}a^{6}-\frac{28}{75}a^{4}+\frac{17}{50}a^{2}-\frac{1}{2}a+\frac{7}{30}$, $\frac{1}{150}a^{11}-\frac{1}{150}a^{9}+\frac{1}{150}a^{7}-\frac{28}{75}a^{5}+\frac{17}{50}a^{3}-\frac{1}{2}a^{2}+\frac{7}{30}a$, $\frac{1}{501750}a^{12}-\frac{7}{20070}a^{10}+\frac{109}{6690}a^{8}-\frac{83}{50175}a^{6}+\frac{9523}{33450}a^{4}-\frac{1}{2}a^{3}+\frac{59861}{501750}a^{2}+\frac{12946}{50175}$, $\frac{1}{501750}a^{13}-\frac{7}{20070}a^{11}+\frac{109}{6690}a^{9}-\frac{83}{50175}a^{7}+\frac{9523}{33450}a^{5}-\frac{1}{2}a^{4}+\frac{59861}{501750}a^{3}+\frac{12946}{50175}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$, $5$ |
Class group and class number
$C_{2}$, which has order $2$
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: |
$\frac{71}{250875}a^{12}+\frac{191}{50175}a^{10}-\frac{107}{16725}a^{8}-\frac{484}{10035}a^{6}+\frac{2896}{16725}a^{4}-\frac{1364}{250875}a^{2}+\frac{25342}{50175}$, $\frac{4}{250875}a^{13}+\frac{241}{250875}a^{12}+\frac{389}{100350}a^{11}+\frac{262}{50175}a^{10}+\frac{132}{5575}a^{9}-\frac{2903}{33450}a^{8}-\frac{15382}{50175}a^{7}+\frac{18866}{50175}a^{6}+\frac{6156}{5575}a^{5}-\frac{11446}{16725}a^{4}-\frac{655067}{501750}a^{3}+\frac{524197}{501750}a^{2}+\frac{40013}{50175}a-\frac{45283}{50175}$, $\frac{577}{501750}a^{13}+\frac{43}{250875}a^{12}+\frac{272}{50175}a^{11}+\frac{67}{20070}a^{10}-\frac{3533}{33450}a^{9}+\frac{4}{3345}a^{8}+\frac{27706}{50175}a^{7}-\frac{10931}{100350}a^{6}-\frac{43657}{33450}a^{5}+\frac{23983}{33450}a^{4}+\frac{446221}{250875}a^{3}-\frac{285952}{250875}a^{2}-\frac{89401}{100350}a+\frac{102637}{100350}$, $\frac{241}{83625}a^{13}-\frac{88}{83625}a^{12}+\frac{262}{16725}a^{11}-\frac{14}{5575}a^{10}-\frac{2903}{11150}a^{9}+\frac{654}{5575}a^{8}+\frac{18866}{16725}a^{7}-\frac{7581}{11150}a^{6}-\frac{11446}{5575}a^{5}+\frac{4912}{3345}a^{4}+\frac{220286}{83625}a^{3}-\frac{49601}{27875}a^{2}-\frac{40391}{33450}a+\frac{19331}{11150}$, $\frac{577}{501750}a^{13}-\frac{43}{250875}a^{12}+\frac{272}{50175}a^{11}-\frac{67}{20070}a^{10}-\frac{3533}{33450}a^{9}-\frac{4}{3345}a^{8}+\frac{27706}{50175}a^{7}+\frac{10931}{100350}a^{6}-\frac{43657}{33450}a^{5}-\frac{23983}{33450}a^{4}+\frac{446221}{250875}a^{3}+\frac{285952}{250875}a^{2}-\frac{89401}{100350}a-\frac{102637}{100350}$, $\frac{4}{250875}a^{13}-\frac{241}{250875}a^{12}+\frac{389}{100350}a^{11}-\frac{262}{50175}a^{10}+\frac{132}{5575}a^{9}+\frac{2903}{33450}a^{8}-\frac{15382}{50175}a^{7}-\frac{18866}{50175}a^{6}+\frac{6156}{5575}a^{5}+\frac{11446}{16725}a^{4}-\frac{655067}{501750}a^{3}-\frac{524197}{501750}a^{2}+\frac{40013}{50175}a+\frac{45283}{50175}$
| 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: | \( 12364.0517956 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 12364.0517956 \cdot 2}{2\cdot\sqrt{12545167784987734375}}\cr\approx \mathstrut & 1.34952724631 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 + 6*x^12 - 85*x^10 + 355*x^8 - 695*x^6 + 1091*x^4 - 609*x^2 + 535)
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 + 6*x^12 - 85*x^10 + 355*x^8 - 695*x^6 + 1091*x^4 - 609*x^2 + 535, 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 + 6*x^12 - 85*x^10 + 355*x^8 - 695*x^6 + 1091*x^4 - 609*x^2 + 535);
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 + 6*x^12 - 85*x^10 + 355*x^8 - 695*x^6 + 1091*x^4 - 609*x^2 + 535);
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 14 |
| The 5 conjugacy class representatives for $D_{7}$ |
| Character table for $D_{7}$ |
Intermediate fields
| \(\Q(\sqrt{-535}) \), 7.1.153130375.1 x7 |
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 sibling: | 7.1.153130375.1 |
| Minimal sibling: | 7.1.153130375.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{7}$ | R | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{7}$ | ${\href{/padicField/37.2.0.1}{2} }^{7}$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(107\)
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.535.2t1.a.a | $1$ | $ 5 \cdot 107 $ | \(\Q(\sqrt{-535}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *2 | 2.535.7t2.b.c | $2$ | $ 5 \cdot 107 $ | 14.0.12545167784987734375.1 | $D_{7}$ (as 14T2) | $1$ | $0$ |
| *2 | 2.535.7t2.b.a | $2$ | $ 5 \cdot 107 $ | 14.0.12545167784987734375.1 | $D_{7}$ (as 14T2) | $1$ | $0$ |
| *2 | 2.535.7t2.b.b | $2$ | $ 5 \cdot 107 $ | 14.0.12545167784987734375.1 | $D_{7}$ (as 14T2) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.