Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 25*x^24 + 24*x^23 + 276*x^22 - 253*x^21 - 1771*x^20 + 1540*x^19 + 7315*x^18 - 5985*x^17 - 20349*x^16 + 15504*x^15 + 38760*x^14 - 27132*x^13 - 50388*x^12 + 31824*x^11 + 43758*x^10 - 24310*x^9 - 24310*x^8 + 11440*x^7 + 8008*x^6 - 3003*x^5 - 1365*x^4 + 364*x^3 + 91*x^2 - 13*x - 1)
gp: K = bnfinit(x^26 - x^25 - 25*x^24 + 24*x^23 + 276*x^22 - 253*x^21 - 1771*x^20 + 1540*x^19 + 7315*x^18 - 5985*x^17 - 20349*x^16 + 15504*x^15 + 38760*x^14 - 27132*x^13 - 50388*x^12 + 31824*x^11 + 43758*x^10 - 24310*x^9 - 24310*x^8 + 11440*x^7 + 8008*x^6 - 3003*x^5 - 1365*x^4 + 364*x^3 + 91*x^2 - 13*x - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -13, 91, 364, -1365, -3003, 8008, 11440, -24310, -24310, 43758, 31824, -50388, -27132, 38760, 15504, -20349, -5985, 7315, 1540, -1771, -253, 276, 24, -25, -1, 1]);
\( x^{26} - x^{25} - 25 x^{24} + 24 x^{23} + 276 x^{22} - 253 x^{21} - 1771 x^{20} + 1540 x^{19} + 7315 x^{18} - 5985 x^{17} - 20349 x^{16} + 15504 x^{15} + 38760 x^{14} - 27132 x^{13} - 50388 x^{12} + 31824 x^{11} + 43758 x^{10} - 24310 x^{9} - 24310 x^{8} + 11440 x^{7} + 8008 x^{6} - 3003 x^{5} - 1365 x^{4} + 364 x^{3} + 91 x^{2} - 13 x - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[26, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(12790771483610519443342791266451996229460693\)\(\medspace = 53^{25}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $45.49$ | 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: | $53$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $26$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{53}(1,·)$, $\chi_{53}(4,·)$, $\chi_{53}(6,·)$, $\chi_{53}(7,·)$, $\chi_{53}(9,·)$, $\chi_{53}(10,·)$, $\chi_{53}(11,·)$, $\chi_{53}(13,·)$, $\chi_{53}(15,·)$, $\chi_{53}(16,·)$, $\chi_{53}(17,·)$, $\chi_{53}(24,·)$, $\chi_{53}(25,·)$, $\chi_{53}(28,·)$, $\chi_{53}(29,·)$, $\chi_{53}(36,·)$, $\chi_{53}(37,·)$, $\chi_{53}(38,·)$, $\chi_{53}(40,·)$, $\chi_{53}(42,·)$, $\chi_{53}(43,·)$, $\chi_{53}(44,·)$, $\chi_{53}(46,·)$, $\chi_{53}(47,·)$, $\chi_{53}(49,·)$, $\chi_{53}(52,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$
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: | $25$ | 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: | \( 14915851505236.459 \) (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 cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{53}) \), 13.13.491258904256726154641.1 |
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 | $26$ | $26$ | $26$ | ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ | R | ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 53 | Data not computed | ||||||