Properties

Label 4.4.3048625.1
Degree $4$
Signature $[4, 0]$
Discriminant $3048625$
Root discriminant $41.79$
Ramified primes $5, 29$
Class number $4$
Class group $[4]$
Galois group $C_4$ (as 4T1)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 54*x^2 - 136*x - 64)
 
gp: K = bnfinit(x^4 - x^3 - 54*x^2 - 136*x - 64, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -136, -54, -1, 1]);
 

\(x^{4} - x^{3} - 54 x^{2} - 136 x - 64\)  Toggle raw display

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

Invariants

Degree:  $4$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[4, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3048625\)\(\medspace = 5^{3}\cdot 29^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $41.79$
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:  $5, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $4$
This field is Galois and abelian over $\Q$.
Conductor:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(128,·)$, $\chi_{145}(1,·)$, $\chi_{145}(144,·)$, $\chi_{145}(17,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$  Toggle raw display

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

Class group and class number

$C_{4}$, which has order $4$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - 22 a - 37 \),  \( 3 a^{3} - 5 a^{2} - 160 a - 311 \),  \( 2 a^{2} + 6 a + 3 \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 243.301642293 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{0}\cdot 243.301642293 \cdot 4}{2\sqrt{3048625}}\approx 4.45905691220$

Galois group

$C_4$ (as 4T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 4
The 4 conjugacy class representatives for $C_4$
Character table for $C_4$

Intermediate fields

\(\Q(\sqrt{145}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.1.0.1}{1} }^{4}$ ${\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.4.0.1}{4} }$ R ${\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.2.0.1}{2} }^{2}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.145.4t1.c.a$1$ $ 5 \cdot 29 $ 4.4.3048625.1 $C_4$ (as 4T1) $0$ $1$
* 1.145.2t1.a.a$1$ $ 5 \cdot 29 $ \(\Q(\sqrt{145}) \) $C_2$ (as 2T1) $1$ $1$
* 1.145.4t1.c.b$1$ $ 5 \cdot 29 $ 4.4.3048625.1 $C_4$ (as 4T1) $0$ $1$

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 *.