Properties

Label 7.7.13841287201.1
Degree $7$
Signature $[7, 0]$
Discriminant $13841287201$
Root discriminant \(28.10\)
Ramified prime see page
Class number $1$
Class group trivial
Galois group $C_7$ (as 7T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 21*x^5 - 21*x^4 + 91*x^3 + 112*x^2 - 84*x - 97)
 
gp: K = bnfinit(x^7 - 21*x^5 - 21*x^4 + 91*x^3 + 112*x^2 - 84*x - 97, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-97, -84, 112, 91, -21, -21, 0, 1]);
 

\( x^{7} - 21x^{5} - 21x^{4} + 91x^{3} + 112x^{2} - 84x - 97 \) Copy content Toggle raw display

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

Invariants

Degree:  $7$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[7, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(13841287201\) \(\medspace = 7^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(28.10\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Gal(K/\Q) }$:  $7$
This field is Galois and abelian over $\Q$.
Conductor:  \(49=7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{49}(1,·)$, $\chi_{49}(36,·)$, $\chi_{49}(22,·)$, $\chi_{49}(8,·)$, $\chi_{49}(43,·)$, $\chi_{49}(29,·)$, $\chi_{49}(15,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{589}a^{6}+\frac{239}{589}a^{5}-\frac{33}{589}a^{4}-\frac{251}{589}a^{3}+\frac{180}{589}a^{2}+\frac{135}{589}a-\frac{214}{589}$ Copy content Toggle raw display

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

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);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
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);
 
Fundamental units:   $\frac{125}{589}a^{6}-\frac{164}{589}a^{5}-\frac{2358}{589}a^{4}+\frac{431}{589}a^{3}+\frac{10131}{589}a^{2}+\frac{972}{589}a-\frac{10258}{589}$, $\frac{315}{589}a^{6}-\frac{696}{589}a^{5}-\frac{5094}{589}a^{4}+\frac{4573}{589}a^{3}+\frac{19004}{589}a^{2}-\frac{5773}{589}a-\frac{14400}{589}$, $\frac{110}{589}a^{6}-\frac{215}{589}a^{5}-\frac{1863}{589}a^{4}+\frac{1251}{589}a^{3}+\frac{7431}{589}a^{2}-\frac{1642}{589}a-\frac{5870}{589}$, $\frac{10}{589}a^{6}+\frac{34}{589}a^{5}-\frac{330}{589}a^{4}-\frac{743}{589}a^{3}+\frac{2389}{589}a^{2}+\frac{3117}{589}a-\frac{5085}{589}$, $\frac{320}{589}a^{6}-\frac{679}{589}a^{5}-\frac{5259}{589}a^{4}+\frac{4496}{589}a^{3}+\frac{19315}{589}a^{2}-\frac{6276}{589}a-\frac{14881}{589}$, $\frac{379}{589}a^{6}-\frac{714}{589}a^{5}-\frac{6617}{589}a^{4}+\frac{4412}{589}a^{3}+\frac{26401}{589}a^{2}-\frac{5968}{589}a-\frac{20439}{589}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 550.88577714 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{7}\cdot(2\pi)^{0}\cdot 550.88577714 \cdot 1}{2\sqrt{13841287201}}\approx 0.29967691809$

Galois group

$C_7$ (as 7T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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 $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }$ ${\href{/padicField/3.7.0.1}{7} }$ ${\href{/padicField/5.7.0.1}{7} }$ R ${\href{/padicField/11.7.0.1}{7} }$ ${\href{/padicField/13.7.0.1}{7} }$ ${\href{/padicField/17.7.0.1}{7} }$ ${\href{/padicField/19.1.0.1}{1} }^{7}$ ${\href{/padicField/23.7.0.1}{7} }$ ${\href{/padicField/29.7.0.1}{7} }$ ${\href{/padicField/31.1.0.1}{1} }^{7}$ ${\href{/padicField/37.7.0.1}{7} }$ ${\href{/padicField/41.7.0.1}{7} }$ ${\href{/padicField/43.7.0.1}{7} }$ ${\href{/padicField/47.7.0.1}{7} }$ ${\href{/padicField/53.7.0.1}{7} }$ ${\href{/padicField/59.7.0.1}{7} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.49.7t1.a.a$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
* 1.49.7t1.a.b$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
* 1.49.7t1.a.c$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
* 1.49.7t1.a.d$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
* 1.49.7t1.a.e$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
* 1.49.7t1.a.f$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.