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Properties

Label 7.7.316838611601521.1
Degree $7$
Signature $[7, 0]$
Discriminant $3.168\times 10^{14}$
Root discriminant \(117.91\)
Ramified prime $4219$
Class number $1$
Class group trivial
Galois group $C_7:C_3$ (as 7T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 45*x^5 - 37*x^4 + 267*x^3 + 59*x^2 - 287*x - 80)
 
gp: K = bnfinit(y^7 - y^6 - 45*y^5 - 37*y^4 + 267*y^3 + 59*y^2 - 287*y - 80, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^7 - x^6 - 45*x^5 - 37*x^4 + 267*x^3 + 59*x^2 - 287*x - 80);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^7 - x^6 - 45*x^5 - 37*x^4 + 267*x^3 + 59*x^2 - 287*x - 80)
 

\( x^{7} - x^{6} - 45x^{5} - 37x^{4} + 267x^{3} + 59x^{2} - 287x - 80 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $7$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[7, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(316838611601521\) \(\medspace = 4219^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(117.91\)
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:  $4219^{2/3}\approx 261.09968643070096$
Ramified primes:   \(4219\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{125}a^{6}+\frac{6}{125}a^{5}-\frac{3}{125}a^{4}-\frac{58}{125}a^{3}-\frac{14}{125}a^{2}-\frac{39}{125}a-\frac{12}{25}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  $25$
Inessential primes:  $5$

Class group and class number

Trivial group, which has order $1$

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 $2$) Copy content Toggle raw display
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:   $a^{6}-2a^{5}-43a^{4}+6a^{3}+261a^{2}-202a-83$, $\frac{11}{125}a^{6}-\frac{9}{125}a^{5}-\frac{508}{125}a^{4}-\frac{363}{125}a^{3}+\frac{2746}{125}a^{2}-\frac{1079}{125}a-\frac{107}{25}$, $\frac{117}{125}a^{6}-\frac{673}{125}a^{5}-\frac{2126}{125}a^{4}+\frac{6439}{125}a^{3}+\frac{337}{125}a^{2}-\frac{6088}{125}a+\frac{321}{25}$, $\frac{98}{25}a^{6}-\frac{432}{25}a^{5}-\frac{2894}{25}a^{4}+\frac{6116}{25}a^{3}+\frac{3853}{25}a^{2}-\frac{6927}{25}a-\frac{421}{5}$, $\frac{2}{25}a^{6}-\frac{8}{25}a^{5}-\frac{116}{25}a^{4}-\frac{1}{25}a^{3}+\frac{787}{25}a^{2}-\frac{193}{25}a-\frac{179}{5}$, $\frac{5108}{125}a^{6}-\frac{6602}{125}a^{5}-\frac{227799}{125}a^{4}-\frac{122489}{125}a^{3}+\frac{1394013}{125}a^{2}-\frac{111687}{125}a-\frac{281671}{25}$ Copy content Toggle raw display
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:  \( 961298.251359 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 961298.251359 \cdot 1}{2\cdot\sqrt{316838611601521}}\cr\approx \mathstrut & 3.45636083624 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 45*x^5 - 37*x^4 + 267*x^3 + 59*x^2 - 287*x - 80)
 
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^7 - x^6 - 45*x^5 - 37*x^4 + 267*x^3 + 59*x^2 - 287*x - 80, 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^7 - x^6 - 45*x^5 - 37*x^4 + 267*x^3 + 59*x^2 - 287*x - 80);
 
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^7 - x^6 - 45*x^5 - 37*x^4 + 267*x^3 + 59*x^2 - 287*x - 80);
 
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

$C_7:C_3$ (as 7T3):

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 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 21
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.1.0.1}{1} }^{7}$ ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.7.0.1}{7} }$ ${\href{/padicField/13.7.0.1}{7} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.7.0.1}{7} }$ ${\href{/padicField/23.7.0.1}{7} }$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.7.0.1}{7} }$ ${\href{/padicField/37.7.0.1}{7} }$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.7.0.1}{7} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(4219\) Copy content Toggle raw display $\Q_{4219}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $3$$3$$1$$2$
Deg $3$$3$$1$$2$