Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1)
gp: K = bnfinit(y^42 - y^35 + y^28 - y^21 + y^14 - y^7 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1)
\( x^{42} - x^{35} + x^{28} - x^{21} + x^{14} - x^{7} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 21]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-118181386580595879976868414312001964434038548836769923458287039207\)
\(\medspace = -\,7^{77}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.43\) | 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: | $7^{11/6}\approx 35.4279812935747$ | ||
| Ramified primes: |
\(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(49=7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{49}(1,·)$, $\chi_{49}(2,·)$, $\chi_{49}(3,·)$, $\chi_{49}(4,·)$, $\chi_{49}(5,·)$, $\chi_{49}(6,·)$, $\chi_{49}(8,·)$, $\chi_{49}(9,·)$, $\chi_{49}(10,·)$, $\chi_{49}(11,·)$, $\chi_{49}(12,·)$, $\chi_{49}(13,·)$, $\chi_{49}(15,·)$, $\chi_{49}(16,·)$, $\chi_{49}(17,·)$, $\chi_{49}(18,·)$, $\chi_{49}(19,·)$, $\chi_{49}(20,·)$, $\chi_{49}(22,·)$, $\chi_{49}(23,·)$, $\chi_{49}(24,·)$, $\chi_{49}(25,·)$, $\chi_{49}(26,·)$, $\chi_{49}(27,·)$, $\chi_{49}(29,·)$, $\chi_{49}(30,·)$, $\chi_{49}(31,·)$, $\chi_{49}(32,·)$, $\chi_{49}(33,·)$, $\chi_{49}(34,·)$, $\chi_{49}(36,·)$, $\chi_{49}(37,·)$, $\chi_{49}(38,·)$, $\chi_{49}(39,·)$, $\chi_{49}(40,·)$, $\chi_{49}(41,·)$, $\chi_{49}(43,·)$, $\chi_{49}(44,·)$, $\chi_{49}(45,·)$, $\chi_{49}(46,·)$, $\chi_{49}(47,·)$, $\chi_{49}(48,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{1048576}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{43}$, which has order $43$ (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: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( a \)
(order $98$)
| 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: |
$a^{38}+a^{24}+a^{10}$, $a^{34}-a^{13}$, $a-1$, $a^{2}+1$, $a^{4}+1$, $a^{3}-1$, $a^{2}-a+1$, $a^{6}+1$, $a^{8}+1$, $a^{33}+a^{17}$, $a^{32}-a^{15}$, $a^{31}+a^{13}$, $a^{41}+a^{40}+a^{39}+a^{38}+a^{37}+a^{36}+a^{35}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{13}+a^{12}+a^{10}+a^{8}-a^{5}-a^{3}-a$, $a^{9}-1$, $a^{40}+a^{30}+a^{20}+a^{10}+1$, $a^{5}-1$, $a^{27}+a^{5}$, $a^{34}-a^{19}$, $a^{29}+a^{9}$, $a^{41}-a^{34}+a^{27}+a^{24}-a^{20}+a^{13}-a^{6}+1$
| 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: | \( 1776855897760068.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot 1776855897760068.5 \cdot 43}{98\cdot\sqrt{118181386580595879976868414312001964434038548836769923458287039207}}\cr\approx \mathstrut & 0.131038008164406 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 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^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 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^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 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^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 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 cyclic group of order 42 |
| The 42 conjugacy class representatives for $C_{42}$ |
| Character table for $C_{42}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 7.7.13841287201.1, 14.0.1341068619663964900807.1, \(\Q(\zeta_{49})^+\) |
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 | $21^{2}$ | $42$ | $42$ | R | $21^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{3}$ | $42$ | ${\href{/padicField/19.6.0.1}{6} }^{7}$ | $21^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{7}$ | $21^{2}$ | ${\href{/padicField/41.14.0.1}{14} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{6}$ | $42$ | $21^{2}$ | $42$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| Deg $42$ | $42$ | $1$ | $77$ |