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Properties

Label 7.7.188180785490436649.1
Degree $7$
Signature $[7, 0]$
Discriminant $1.882\times 10^{17}$
Root discriminant \(293.63\)
Ramified prime $757$
Class number $1$
Class group trivial
Galois group $C_7$ (as 7T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 324*x^5 + 1483*x^4 + 20876*x^3 - 129744*x^2 + 36999*x + 54027)
 
gp: K = bnfinit(y^7 - y^6 - 324*y^5 + 1483*y^4 + 20876*y^3 - 129744*y^2 + 36999*y + 54027, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^7 - x^6 - 324*x^5 + 1483*x^4 + 20876*x^3 - 129744*x^2 + 36999*x + 54027);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^7 - x^6 - 324*x^5 + 1483*x^4 + 20876*x^3 - 129744*x^2 + 36999*x + 54027)
 

\( x^{7} - x^{6} - 324x^{5} + 1483x^{4} + 20876x^{3} - 129744x^{2} + 36999x + 54027 \) 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:   \(188180785490436649\) \(\medspace = 757^{6}\) 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:  \(293.63\)
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:  $757^{6/7}\approx 293.6273246706745$
Ramified primes:   \(757\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $7$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(757\)
Dirichlet character group:    $\lbrace$$\chi_{757}(1,·)$, $\chi_{757}(453,·)$, $\chi_{757}(630,·)$, $\chi_{757}(232,·)$, $\chi_{757}(59,·)$, $\chi_{757}(77,·)$, $\chi_{757}(62,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{477}a^{5}-\frac{50}{477}a^{4}-\frac{40}{477}a^{3}+\frac{125}{477}a^{2}-\frac{70}{159}a-\frac{2}{53}$, $\frac{1}{106460199}a^{6}-\frac{39784}{106460199}a^{5}-\frac{5618227}{35486733}a^{4}-\frac{79061}{106460199}a^{3}-\frac{47831770}{106460199}a^{2}-\frac{9769132}{35486733}a+\frac{1578096}{11828911}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  $81$
Inessential primes:  $3$

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:   $\frac{43574327}{106460199}a^{6}+\frac{4101526}{35486733}a^{5}-\frac{256969115}{2008683}a^{4}+\frac{16682286833}{35486733}a^{3}+\frac{839378450828}{106460199}a^{2}-\frac{532153682354}{11828911}a+\frac{395886904931}{11828911}$, $\frac{19550128}{35486733}a^{6}-\frac{1069955870}{106460199}a^{5}-\frac{569905382}{106460199}a^{4}+\frac{97109655968}{106460199}a^{3}-\frac{450035190610}{106460199}a^{2}+\frac{41139713165}{35486733}a+\frac{19808539819}{11828911}$, $\frac{1429201}{11828911}a^{6}+\frac{72855469}{106460199}a^{5}-\frac{3249210818}{106460199}a^{4}+\frac{1686960383}{106460199}a^{3}+\frac{188303941166}{106460199}a^{2}-\frac{246764115811}{35486733}a+\frac{56619230137}{11828911}$, $\frac{53102665}{106460199}a^{6}+\frac{65108860}{106460199}a^{5}-\frac{16607380057}{106460199}a^{4}+\frac{45624212771}{106460199}a^{3}+\frac{121818731631}{11828911}a^{2}-\frac{542845212457}{11828911}a-\frac{303055026751}{11828911}$, $\frac{1333967}{106460199}a^{6}+\frac{9574708}{106460199}a^{5}-\frac{82480060}{35486733}a^{4}-\frac{776893954}{106460199}a^{3}+\frac{12120633712}{106460199}a^{2}-\frac{1367059595}{35486733}a-\frac{579535369}{11828911}$, $\frac{140766}{11828911}a^{6}-\frac{13405093}{106460199}a^{5}-\frac{284851267}{106460199}a^{4}+\frac{4657644829}{106460199}a^{3}-\frac{18155099210}{106460199}a^{2}+\frac{1879962445}{35486733}a+\frac{885381776}{11828911}$ 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:  \( 31817260.5598 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 31817260.5598 \cdot 1}{2\cdot\sqrt{188180785490436649}}\cr\approx \mathstrut & 4.69413007730 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 324*x^5 + 1483*x^4 + 20876*x^3 - 129744*x^2 + 36999*x + 54027)
 
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 - 324*x^5 + 1483*x^4 + 20876*x^3 - 129744*x^2 + 36999*x + 54027, 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 - 324*x^5 + 1483*x^4 + 20876*x^3 - 129744*x^2 + 36999*x + 54027);
 
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 - 324*x^5 + 1483*x^4 + 20876*x^3 - 129744*x^2 + 36999*x + 54027);
 
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

$C_7$ (as 7T1):

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 7
The 7 conjugacy class representatives for $C_7$
Character table for $C_7$

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }$ ${\href{/padicField/3.1.0.1}{1} }^{7}$ ${\href{/padicField/5.7.0.1}{7} }$ ${\href{/padicField/7.7.0.1}{7} }$ ${\href{/padicField/11.7.0.1}{7} }$ ${\href{/padicField/13.7.0.1}{7} }$ ${\href{/padicField/17.7.0.1}{7} }$ ${\href{/padicField/19.7.0.1}{7} }$ ${\href{/padicField/23.1.0.1}{1} }^{7}$ ${\href{/padicField/29.1.0.1}{1} }^{7}$ ${\href{/padicField/31.7.0.1}{7} }$ ${\href{/padicField/37.7.0.1}{7} }$ ${\href{/padicField/41.7.0.1}{7} }$ ${\href{/padicField/43.7.0.1}{7} }$ ${\href{/padicField/47.7.0.1}{7} }$ ${\href{/padicField/53.1.0.1}{1} }^{7}$ ${\href{/padicField/59.7.0.1}{7} }$

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
\(757\) Copy content Toggle raw display Deg $7$$7$$1$$6$