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Properties

Label 7.7.81905390937410041.1
Degree $7$
Signature $[7, 0]$
Discriminant $8.191\times 10^{16}$
Root discriminant \(260.73\)
Ramified prime $659$
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 - 282*x^5 - 1345*x^4 + 5370*x^3 + 30042*x^2 - 14893*x - 115169)
 
gp: K = bnfinit(y^7 - y^6 - 282*y^5 - 1345*y^4 + 5370*y^3 + 30042*y^2 - 14893*y - 115169, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^7 - x^6 - 282*x^5 - 1345*x^4 + 5370*x^3 + 30042*x^2 - 14893*x - 115169);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^7 - x^6 - 282*x^5 - 1345*x^4 + 5370*x^3 + 30042*x^2 - 14893*x - 115169)
 

\( x^{7} - x^{6} - 282x^{5} - 1345x^{4} + 5370x^{3} + 30042x^{2} - 14893x - 115169 \) 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:   \(81905390937410041\) \(\medspace = 659^{6}\) 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:  \(260.73\)
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:  $659^{6/7}\approx 260.7278949189025$
Ramified primes:   \(659\) 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:  \(659\)
Dirichlet character group:    $\lbrace$$\chi_{659}(144,·)$, $\chi_{659}(1,·)$, $\chi_{659}(307,·)$, $\chi_{659}(389,·)$, $\chi_{659}(55,·)$, $\chi_{659}(410,·)$, $\chi_{659}(12,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{53}a^{5}-\frac{24}{53}a^{4}+\frac{5}{53}a^{3}+\frac{24}{53}a^{2}-\frac{5}{53}a$, $\frac{1}{1032220633}a^{6}+\frac{6478640}{1032220633}a^{5}-\frac{268540694}{1032220633}a^{4}+\frac{49142828}{1032220633}a^{3}+\frac{257734100}{1032220633}a^{2}+\frac{336801606}{1032220633}a-\frac{158939}{475021}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  $1$
Inessential primes:  None

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:   $\frac{383623}{1032220633}a^{6}+\frac{1738788}{1032220633}a^{5}-\frac{118763866}{1032220633}a^{4}-\frac{1054424874}{1032220633}a^{3}+\frac{891036565}{1032220633}a^{2}+\frac{26422830093}{1032220633}a+\frac{22415508}{475021}$, $\frac{56332227}{1032220633}a^{6}+\frac{65865963}{1032220633}a^{5}-\frac{15758753143}{1032220633}a^{4}-\frac{109862255963}{1032220633}a^{3}+\frac{68239556769}{1032220633}a^{2}+\frac{34803169714}{19475861}a+\frac{1415451911}{475021}$, $\frac{16374336}{1032220633}a^{6}-\frac{87042663}{1032220633}a^{5}-\frac{4252340873}{1032220633}a^{4}-\frac{3613978205}{1032220633}a^{3}+\frac{106127465247}{1032220633}a^{2}+\frac{40619958890}{1032220633}a-\frac{213520225}{475021}$, $\frac{24023}{25176113}a^{6}+\frac{13259}{25176113}a^{5}-\frac{7371023}{25176113}a^{4}-\frac{38610453}{25176113}a^{3}+\frac{218049521}{25176113}a^{2}+\frac{654368976}{25176113}a-\frac{34511334}{475021}$, $\frac{19179899}{1032220633}a^{6}-\frac{188543118}{1032220633}a^{5}-\frac{4472221920}{1032220633}a^{4}+\frac{15782974786}{1032220633}a^{3}+\frac{147440321668}{1032220633}a^{2}-\frac{41504618308}{1032220633}a-\frac{288337996}{475021}$, $\frac{90413}{24005131}a^{6}-\frac{452481}{24005131}a^{5}-\frac{23432205}{24005131}a^{4}-\frac{29845438}{24005131}a^{3}+\frac{551803727}{24005131}a^{2}+\frac{637550473}{24005131}a-\frac{1407283}{11047}$ 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:  \( 1395957.12953 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 1395957.12953 \cdot 1}{2\cdot\sqrt{81905390937410041}}\cr\approx \mathstrut & 0.312173340220 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 282*x^5 - 1345*x^4 + 5370*x^3 + 30042*x^2 - 14893*x - 115169)
 
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 - 282*x^5 - 1345*x^4 + 5370*x^3 + 30042*x^2 - 14893*x - 115169, 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 - 282*x^5 - 1345*x^4 + 5370*x^3 + 30042*x^2 - 14893*x - 115169);
 
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 - 282*x^5 - 1345*x^4 + 5370*x^3 + 30042*x^2 - 14893*x - 115169);
 
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 ${\href{/padicField/2.7.0.1}{7} }$ ${\href{/padicField/3.7.0.1}{7} }$ ${\href{/padicField/5.7.0.1}{7} }$ ${\href{/padicField/7.7.0.1}{7} }$ ${\href{/padicField/11.7.0.1}{7} }$ ${\href{/padicField/13.7.0.1}{7} }$ ${\href{/padicField/17.7.0.1}{7} }$ ${\href{/padicField/19.7.0.1}{7} }$ ${\href{/padicField/23.7.0.1}{7} }$ ${\href{/padicField/29.7.0.1}{7} }$ ${\href{/padicField/31.1.0.1}{1} }^{7}$ ${\href{/padicField/37.7.0.1}{7} }$ ${\href{/padicField/41.1.0.1}{1} }^{7}$ ${\href{/padicField/43.1.0.1}{1} }^{7}$ ${\href{/padicField/47.1.0.1}{1} }^{7}$ ${\href{/padicField/53.1.0.1}{1} }^{7}$ ${\href{/padicField/59.7.0.1}{7} }$

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
\(659\) Copy content Toggle raw display Deg $7$$7$$1$$6$