# Properties

 Label 22.12.257...080.1 Degree $22$ Signature $[12, 5]$ Discriminant $-2.571\times 10^{72}$ Root discriminant $1956.01$ Ramified primes $2, 3, 5, 7, 13, 19, 23, 137$ Class number $1$ (GRH) Class group trivial (GRH) Galois group 22T52

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 25*x^20 + 244*x^18 - 1152*x^16 + 2229*x^14 + 3057*x^12 - 27810*x^10 + 70572*x^8 - 95244*x^6 + 70846*x^4 - 25252*x^2 + 2470)

gp: K = bnfinit(x^22 - 25*x^20 + 244*x^18 - 1152*x^16 + 2229*x^14 + 3057*x^12 - 27810*x^10 + 70572*x^8 - 95244*x^6 + 70846*x^4 - 25252*x^2 + 2470, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2470, 0, -25252, 0, 70846, 0, -95244, 0, 70572, 0, -27810, 0, 3057, 0, 2229, 0, -1152, 0, 244, 0, -25, 0, 1]);

$$x^{22} - 25 x^{20} + 244 x^{18} - 1152 x^{16} + 2229 x^{14} + 3057 x^{12} - 27810 x^{10} + 70572 x^{8} - 95244 x^{6} + 70846 x^{4} - 25252 x^{2} + 2470$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $22$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[12, 5]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-25\!\cdots\!080$$$$\medspace = -\,2^{43}\cdot 3^{28}\cdot 5\cdot 7^{4}\cdot 13\cdot 19\cdot 23^{4}\cdot 137^{16}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $1956.01$ 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: $2, 3, 5, 7, 13, 19, 23, 137$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13}$, $\frac{1}{12} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{6}$, $\frac{1}{12} a^{19} - \frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{6} a$, $\frac{1}{27000} a^{20} + \frac{241}{6750} a^{18} - \frac{9}{50} a^{16} + \frac{164}{375} a^{14} + \frac{947}{9000} a^{12} - \frac{161}{500} a^{10} - \frac{61}{125} a^{8} - \frac{41}{2250} a^{6} + \frac{169}{375} a^{4} + \frac{4499}{13500} a^{2} - \frac{923}{2700}$, $\frac{1}{54000} a^{21} - \frac{1}{54000} a^{20} + \frac{241}{13500} a^{19} - \frac{241}{13500} a^{18} + \frac{4}{25} a^{17} - \frac{4}{25} a^{16} + \frac{82}{375} a^{15} - \frac{82}{375} a^{14} - \frac{3553}{18000} a^{13} + \frac{3553}{18000} a^{12} - \frac{161}{1000} a^{11} + \frac{161}{1000} a^{10} + \frac{32}{125} a^{9} - \frac{32}{125} a^{8} + \frac{2209}{4500} a^{7} - \frac{2209}{4500} a^{6} - \frac{103}{375} a^{5} + \frac{103}{375} a^{4} - \frac{9001}{27000} a^{3} + \frac{9001}{27000} a^{2} - \frac{923}{5400} a + \frac{923}{5400}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $16$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$642145413743000000000000000000$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{12}\cdot(2\pi)^{5}\cdot 642145413743000000000000000000 \cdot 1}{2\sqrt{2571487881141851171102880299437945407668884201494634232459550622042030080}}\approx 8.03101502217204$ (assuming GRH)

## Galois group

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 40874803200 The 400 conjugacy class representatives for t22n52 are not computed Character table for t22n52 is not computed

## Intermediate fields

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

## Sibling fields

 Degree 44 sibling: data not computed

## 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 R R ${\href{/padicField/11.11.0.1}{11} }^{2}$ R $22$ R R ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

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
$2$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] \Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.8.18.53x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.25.67$x^{12} + 8 x^{10} + 4 x^{8} + 6 x^{6} + 8 x^{4} - 6 x^{2} - 6$$12$$1$$2512T224[4/3, 4/3, 8/3, 8/3, 3, 3, 19/6, 19/6]_{3}^{2} 3Data not computed 55.2.1.2x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 5.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 5.3.0.1x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3} 5.4.0.1x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 77.4.2.1x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 7.7.0.1x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7} 13Data not computed 1919.2.1.1x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5} 19.5.0.1x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10} 2323.2.1.2x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2} 23.4.2.1x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.14.0.1$x^{14} - x + 7$$1$$14$$0$$C_{14}$$[\ ]^{14} 137137.2.0.1x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.5.4.1$x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4} 137.5.4.1x^{5} - 137$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
137.10.8.1$x^{10} - 137 x^{5} + 112614$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$