Degree $1$
Signature $[1, 0]$
Discriminant $1$
Root discriminant \(1.00\)
Ramified primes $$
Class number $1$
Class group trivial
Galois group Trivial (as 1T1)

Related objects


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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x)
gp: K = bnfinit(x, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x)

\( x \) Copy content Toggle raw display

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


Degree:  $1$
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature:  $[1, 0]$
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant:   \(1\) 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:  \(1.00\)
sage: (K.disc().abs())^(1./
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
Ramified primes:   \textrm{None} 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $1$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:  \(1\)
Dirichlet character group:    $\lbrace$$\chi_{1}(1,·)$$\rbrace$
This is not a CM field.

Integral basis

$1$ Copy content Toggle raw display

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

Monogenic:  Yes
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:  $0$
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)
Regulator:  \( 1 \)
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} =\frac{2^1 (2\pi)^0 \cdot 1\cdot 1}{2\cdot\sqrt 1}=1$

# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x)
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, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * / (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);
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);
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

$C_1$ (as 1T1):

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 1
The conjugacy class representative for Trivial
Character table for Trivial

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 ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$ ${\href{/padicField/}{1} }$

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

There are no ramified primes.

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$

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 *.