LMFDB - Number field 2.0.225032808.1: $\Q(\sqrt{-56258202}) $

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^2 + 56258202)

gp:K = bnfinit(y^2 + 56258202, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^2 + 56258202);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^2 + 56258202)

$ x^{2} + 56258202 $ x^2 + 56258202

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$2$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[0, 1]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$-225032808$ -225032808 $\medspace = -\,2^{3}\cdot 3\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 29$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$$15\,001$.09$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	$2^{3/2}3^{1/2}7^{1/2}11^{1/2}13^{1/2}17^{1/2}19^{1/2}29^{1/2}\approx 15001.09356013754$
Ramified primes:	$2$, $3$, $7$, $11$, $13$, $17$, $19$, $29$ 2, 3, 7, 11, 13, 17, 19, 29	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant(OK))
Discriminant root field:	$\Q(\sqrt{-56258202}) $
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:	$C_2$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$225032808=2^{3}\cdot 3\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 29$
Dirichlet character group:	$\lbrace$$\chi_{225032808}(1,·)$, $\chi_{225032808}(112516403,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{1}$

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

$1$, $a$ 1, a

comment:Integral basis

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

Ideal class group:	$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{54}$, which has order $3456$	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:	$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{54}$, which has order $3456$	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);
Relative class number:	$3456$

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$0$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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 $	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr =\mathstrut &\frac{2^{0}\cdot(2\pi)^{1}\cdot 1 \cdot 3456}{2\cdot\sqrt{225032808}}\cr\approx \mathstrut & 0.723770181639130 \end{aligned}\]

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^2 + 56258202) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^2 + 56258202, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^2 + 56258202); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^2 + 56258202); 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

$C_2$ (as 2T1):

comment:Galois group

sage:K.galois_group(type='pari')

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A cyclic group of order 2

The 2 conjugacy class representatives for $C_2$

Character table for $C_2$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(L)]

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	R	R	${\href{/padicField/5.2.0.1}{2} }$	R	R	R	R	R	${\href{/padicField/23.2.0.1}{2} }$	R	${\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.2.0.1}{2} }$	${\href{/padicField/41.2.0.1}{2} }$	${\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.2.0.1}{2} }$	${\href{/padicField/53.2.0.1}{2} }$	${\href{/padicField/59.1.0.1}{1} }^{2}$

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

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.1.2.3a1.2	$x^{2} + 10$	$2$	$1$	$3$	$C_2$	$$[3]$$
$3$ 3	3.1.2.1a1.1	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
$7$ 7	7.1.2.1a1.1	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
$11$ 11	11.1.2.1a1.1	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
$13$ 13	13.1.2.1a1.1	$x^{2} + 13$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
$17$ 17	17.1.2.1a1.1	$x^{2} + 17$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
$19$ 19	19.1.2.1a1.1	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
$29$ 29	29.1.2.1a1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$

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Normalized defining polynomial

Invariants

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

Class group and class number

Unit group

Class number formula

Galois group

Intermediate fields

Frobenius cycle types

Local algebras for ramified primes

Spectrum of ring of integers

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2.0.225032808.1

Degree	$2$
Signature	$[0, 1]$
Discriminant	$-225032808$
Root discriminant	\($15\,001$.09\)
Ramified primes	$2,3,7,11,13,17,19,29$
Class number	$3456$
Class group	[2, 2, 2, 2, 2, 2, 54]
Galois group	$C_2$ (as 2T1)

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(2\) 2	2.1.2.3a1.2	$x^{2} + 10$	$2$	$1$	$3$	$C_2$	$$[3]$$
\(3\) 3	3.1.2.1a1.1	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(7\) 7	7.1.2.1a1.1	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(11\) 11	11.1.2.1a1.1	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(13\) 13	13.1.2.1a1.1	$x^{2} + 13$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(17\) 17	17.1.2.1a1.1	$x^{2} + 17$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(19\) 19	19.1.2.1a1.1	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(29\) 29	29.1.2.1a1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$