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Properties

Label 7.7.8233120419813614521.3
Degree $7$
Signature $[7, 0]$
Discriminant $8.233\times 10^{18}$
Root discriminant \(503.76\)
Ramified primes $7,29$
Class number $7$
Class group [7]
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 - 609*x^5 - 2233*x^4 + 48111*x^3 - 40194*x^2 - 87696*x + 77517)
 
gp: K = bnfinit(y^7 - 609*y^5 - 2233*y^4 + 48111*y^3 - 40194*y^2 - 87696*y + 77517, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^7 - 609*x^5 - 2233*x^4 + 48111*x^3 - 40194*x^2 - 87696*x + 77517);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^7 - 609*x^5 - 2233*x^4 + 48111*x^3 - 40194*x^2 - 87696*x + 77517)
 

\( x^{7} - 609x^{5} - 2233x^{4} + 48111x^{3} - 40194x^{2} - 87696x + 77517 \) 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:   \(8233120419813614521\) \(\medspace = 7^{12}\cdot 29^{6}\) 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:  \(503.76\)
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:  $7^{12/7}29^{6/7}\approx 503.75978549322235$
Ramified primes:   \(7\), \(29\) 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:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1408,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(645,·)$, $\chi_{1421}(1009,·)$, $\chi_{1421}(169,·)$, $\chi_{1421}(141,·)$, $\chi_{1421}(1093,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{3}-\frac{1}{9}a^{2}$, $\frac{1}{27}a^{4}-\frac{1}{27}a^{3}-\frac{1}{3}a$, $\frac{1}{2673}a^{5}-\frac{1}{243}a^{4}+\frac{10}{2673}a^{3}+\frac{2}{27}a^{2}+\frac{113}{297}a+\frac{1}{3}$, $\frac{1}{3440151}a^{6}+\frac{478}{3440151}a^{5}+\frac{36112}{3440151}a^{4}+\frac{43111}{1146717}a^{3}+\frac{49976}{382239}a^{2}+\frac{401}{127413}a+\frac{586}{1287}$ 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

$C_{7}$, which has order $7$

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:   $\frac{73615}{3440151}a^{6}-\frac{213539}{3440151}a^{5}-\frac{52029398}{3440151}a^{4}-\frac{71655908}{1146717}a^{3}+\frac{430333979}{382239}a^{2}-\frac{587176}{127413}a-\frac{2776622}{1287}$, $\frac{1397270}{3440151}a^{6}+\frac{35742539}{3440151}a^{5}+\frac{30300704}{3440151}a^{4}-\frac{902246905}{1146717}a^{3}+\frac{277063387}{382239}a^{2}+\frac{185489257}{127413}a-\frac{1703035}{1287}$, $\frac{273825632}{3440151}a^{6}-\frac{5064132157}{3440151}a^{5}-\frac{73104728194}{3440151}a^{4}+\frac{246854623307}{1146717}a^{3}-\frac{57965025890}{382239}a^{2}-\frac{50390038937}{127413}a+\frac{429474164}{1287}$, $\frac{1129933}{3440151}a^{6}+\frac{970804}{3440151}a^{5}-\frac{686558039}{3440151}a^{4}-\frac{1032302729}{1146717}a^{3}+\frac{5739139610}{382239}a^{2}-\frac{90474616}{127413}a-\frac{35553116}{1287}$, $\frac{20792}{382239}a^{6}-\frac{62987}{382239}a^{5}-\frac{11538914}{382239}a^{4}-\frac{24164524}{382239}a^{3}+\frac{25609751}{14157}a^{2}-\frac{1498280}{42471}a-\frac{1508401}{429}$, $\frac{739663}{3440151}a^{6}-\frac{3644075}{3440151}a^{5}-\frac{449338517}{3440151}a^{4}+\frac{186320686}{1146717}a^{3}+\frac{4785871709}{382239}a^{2}-\frac{7512413497}{127413}a+\frac{54716170}{1287}$ 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:  \( 64309241.25 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 64309241.25 \cdot 7}{2\cdot\sqrt{8233120419813614521}}\cr\approx \mathstrut & 10.04081963 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^7 - 609*x^5 - 2233*x^4 + 48111*x^3 - 40194*x^2 - 87696*x + 77517)
 
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 - 609*x^5 - 2233*x^4 + 48111*x^3 - 40194*x^2 - 87696*x + 77517, 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 - 609*x^5 - 2233*x^4 + 48111*x^3 - 40194*x^2 - 87696*x + 77517);
 
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 - 609*x^5 - 2233*x^4 + 48111*x^3 - 40194*x^2 - 87696*x + 77517);
 
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 ${\href{/padicField/2.7.0.1}{7} }$ ${\href{/padicField/3.1.0.1}{1} }^{7}$ ${\href{/padicField/5.7.0.1}{7} }$ R ${\href{/padicField/11.1.0.1}{1} }^{7}$ ${\href{/padicField/13.1.0.1}{1} }^{7}$ ${\href{/padicField/17.7.0.1}{7} }$ ${\href{/padicField/19.7.0.1}{7} }$ ${\href{/padicField/23.7.0.1}{7} }$ R ${\href{/padicField/31.7.0.1}{7} }$ ${\href{/padicField/37.7.0.1}{7} }$ ${\href{/padicField/41.7.0.1}{7} }$ ${\href{/padicField/43.7.0.1}{7} }$ ${\href{/padicField/47.1.0.1}{1} }^{7}$ ${\href{/padicField/53.1.0.1}{1} }^{7}$ ${\href{/padicField/59.7.0.1}{7} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.7.12.7$x^{7} + 42 x^{6} + 301$$7$$1$$12$$C_7$$[2]$
\(29\) Copy content Toggle raw display 29.7.6.5$x^{7} + 58$$7$$1$$6$$C_7$$[\ ]_{7}$