Properties

Label 4.2.794983.1
Degree 44
Signature [2,1][2, 1]
Discriminant 794983-794983
Root discriminant 29.8629.86
Ramified primes 7,3377,337
Class number 11
Class group trivial
Galois group D4D_{4} (as 4T3)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 + 15*x^2 - 28)
 
gp: K = bnfinit(y^4 + 15*y^2 - 28, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^4 + 15*x^2 - 28);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^4 + 15*x^2 - 28)
 

x4+15x228 x^{4} + 15x^{2} - 28 Copy content Toggle raw display

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

Invariants

Degree:  44
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [2,1][2, 1]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   794983-794983 =73372\medspace = -\,7\cdot 337^{2} 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:  29.8629.86
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:  71/23371/248.5695377783235647^{1/2}337^{1/2}\approx 48.569537778323564
Ramified primes:   77, 337337 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(7)\Q(\sqrt{-7})
Aut(K/Q)\Aut(K/\Q):   C2C_2
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.

Integral basis (with respect to field generator aa)

11, aa, 12a212a\frac{1}{2}a^{2}-\frac{1}{2}a, 14a3+14a12\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2} Copy content Toggle raw display

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

Monogenic:  No
Index:  22
Inessential primes:  22

Class group and class number

Trivial group, which has order 11

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:  22
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   1 -1  (order 22) 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:   34a3a2+514a332\frac{3}{4}a^{3}-a^{2}+\frac{51}{4}a-\frac{33}{2}, 15414a31885a225874a+63292\frac{1541}{4}a^{3}-1885a^{2}-\frac{2587}{4}a+\frac{6329}{2} 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:  76.9979813664 76.9979813664
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(22(2π)176.997981366412794983(1.08520124340 \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^{2}\cdot(2\pi)^{1}\cdot 76.9979813664 \cdot 1}{2\cdot\sqrt{794983}}\cr\approx \mathstrut & 1.08520124340 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^4 + 15*x^2 - 28)
 
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^4 + 15*x^2 - 28, 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(Rationals()); K<a> := NumberField(x^4 + 15*x^2 - 28);
 
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^4 + 15*x^2 - 28);
 
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

D4D_4 (as 4T3):

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 solvable group of order 8
The 5 conjugacy class representatives for D4D_{4}
Character table for D4D_{4}

Intermediate fields

Q(337)\Q(\sqrt{337})

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

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]
 

Sibling fields

Galois closure: 8.0.30967900544161.1
Degree 4 sibling: 4.0.16513.1
Minimal sibling: 4.0.16513.1

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type 22{\href{/padicField/2.2.0.1}{2} }^{2} 2,12{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2} 4{\href{/padicField/5.4.0.1}{4} } R 22{\href{/padicField/11.2.0.1}{2} }^{2} 2,12{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2} 4{\href{/padicField/17.4.0.1}{4} } 4{\href{/padicField/19.4.0.1}{4} } 22{\href{/padicField/23.2.0.1}{2} }^{2} 22{\href{/padicField/29.2.0.1}{2} }^{2} 4{\href{/padicField/31.4.0.1}{4} } 14{\href{/padicField/37.1.0.1}{1} }^{4} 2,12{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2} 14{\href{/padicField/43.1.0.1}{1} }^{4} 2,12{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2} 22{\href{/padicField/53.2.0.1}{2} }^{2} 4{\href{/padicField/59.4.0.1}{4} }

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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=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

ppLabelPolynomial ee ff cc Galois group Slope content
77 Copy content Toggle raw display 7.1.2.1a1.2x2+21x^{2} + 21221111C2C_2[ ]2[\ ]_{2}
7.2.1.0a1.1x2+6x+3x^{2} + 6 x + 3112200C2C_2[ ]2[\ ]^{2}
337337 Copy content Toggle raw display Deg 44222222

Artin representations

Label Dimension Conductor Artin stem field GG Ind χ(c)\chi(c)
* 1.1.1t1.a.a11 11 Q\Q C1C_1 11 11
* 1.337.2t1.a.a11 337 337 Q(337)\Q(\sqrt{337}) C2C_2 (as 2T1) 11 11
1.7.2t1.a.a11 7 7 Q(7)\Q(\sqrt{-7}) C2C_2 (as 2T1) 11 1-1
1.2359.2t1.a.a11 7337 7 \cdot 337 Q(2359)\Q(\sqrt{-2359}) C2C_2 (as 2T1) 11 1-1
* 2.2359.4t3.a.a22 7337 7 \cdot 337 4.2.794983.1 D4D_{4} (as 4T3) 11 00

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 *.

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)