Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*x + 1)
gp: K = bnfinit(x^22 - 11*x^21 + 64*x^20 - 255*x^19 + 770*x^18 - 1857*x^17 + 3688*x^16 - 6146*x^15 + 8690*x^14 - 10480*x^13 + 10777*x^12 - 9386*x^11 + 6807*x^10 - 3960*x^9 + 1680*x^8 - 341*x^7 - 170*x^6 + 212*x^5 - 109*x^4 + 28*x^3 + 3*x^2 - 5*x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 3, 28, -109, 212, -170, -341, 1680, -3960, 6807, -9386, 10777, -10480, 8690, -6146, 3688, -1857, 770, -255, 64, -11, 1]);
\( x^{22} - 11 x^{21} + 64 x^{20} - 255 x^{19} + 770 x^{18} - 1857 x^{17} + 3688 x^{16} - 6146 x^{15} + 8690 x^{14} - 10480 x^{13} + 10777 x^{12} - 9386 x^{11} + 6807 x^{10} - 3960 x^{9} + 1680 x^{8} - 341 x^{7} - 170 x^{6} + 212 x^{5} - 109 x^{4} + 28 x^{3} + 3 x^{2} - 5 x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(440358787157109245022232573\)\(\medspace = 37\cdot 163^{2}\cdot 14281\cdot 177106931^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $16.26$ | 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: | $37, 163, 14281, 177106931$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
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: | $13$ | 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: | \( 42752.2517797 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A non-solvable group of order 81749606400 |
| The 752 conjugacy class representatives for t22n53 are not computed |
| Character table for t22n53 is not computed |
Intermediate fields
| 11.3.28868429753.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ | $18{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | $16{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $37$ | $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.6.0.1 | $x^{6} - x + 20$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 37.6.0.1 | $x^{6} - x + 20$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $163$ | 163.2.1.2 | $x^{2} + 652$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 163.2.1.2 | $x^{2} + 652$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 163.4.0.1 | $x^{4} - x + 42$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 163.4.0.1 | $x^{4} - x + 42$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 163.10.0.1 | $x^{10} - 2 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 14281 | Data not computed | ||||||
| 177106931 | Data not computed | ||||||