sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 44*x^42 + 901*x^40 - 11400*x^38 + 99790*x^36 - 641208*x^34 + 3131721*x^32 - 11878176*x^30 + 35442612*x^28 - 83778736*x^26 + 157236844*x^24 - 233880352*x^22 + 274130056*x^20 - 250699168*x^18 + 176290339*x^16 - 93382192*x^14 + 36217051*x^12 - 9883588*x^10 + 1792219*x^8 - 197912*x^6 + 11506*x^4 - 264*x^2 + 1)
gp: K = bnfinit(x^44 - 44*x^42 + 901*x^40 - 11400*x^38 + 99790*x^36 - 641208*x^34 + 3131721*x^32 - 11878176*x^30 + 35442612*x^28 - 83778736*x^26 + 157236844*x^24 - 233880352*x^22 + 274130056*x^20 - 250699168*x^18 + 176290339*x^16 - 93382192*x^14 + 36217051*x^12 - 9883588*x^10 + 1792219*x^8 - 197912*x^6 + 11506*x^4 - 264*x^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -264, 0, 11506, 0, -197912, 0, 1792219, 0, -9883588, 0, 36217051, 0, -93382192, 0, 176290339, 0, -250699168, 0, 274130056, 0, -233880352, 0, 157236844, 0, -83778736, 0, 35442612, 0, -11878176, 0, 3131721, 0, -641208, 0, 99790, 0, -11400, 0, 901, 0, -44, 0, 1]);
\( x^{44} - 44 x^{42} + 901 x^{40} - 11400 x^{38} + 99790 x^{36} - 641208 x^{34} + 3131721 x^{32} - 11878176 x^{30} + 35442612 x^{28} - 83778736 x^{26} + 157236844 x^{24} - 233880352 x^{22} + 274130056 x^{20} - 250699168 x^{18} + 176290339 x^{16} - 93382192 x^{14} + 36217051 x^{12} - 9883588 x^{10} + 1792219 x^{8} - 197912 x^{6} + 11506 x^{4} - 264 x^{2} + 1 \)
sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
| Degree: | | $44$ |
|
| Signature: | | $[44, 0]$ |
|
| Discriminant: | | \(482\!\cdots\!224\)\(\medspace = 2^{88}\cdot 23^{42}\) |
magma: Discriminant(Integers(K)); |
| Root discriminant: | | $79.78$ | 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: | | $2, 23$ |
gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(Integers(K))); |
| $|\Gal(K/\Q)|$: | | $44$ |
| This field is Galois and abelian over $\Q$. |
| Conductor: | | \(184=2^{3}\cdot 23\) |
| Dirichlet character group:
| |
$\lbrace$$\chi_{184}(1,·)$, $\chi_{184}(43,·)$, $\chi_{184}(133,·)$, $\chi_{184}(7,·)$, $\chi_{184}(9,·)$, $\chi_{184}(11,·)$, $\chi_{184}(13,·)$, $\chi_{184}(15,·)$, $\chi_{184}(19,·)$, $\chi_{184}(25,·)$, $\chi_{184}(155,·)$, $\chi_{184}(29,·)$, $\chi_{184}(159,·)$, $\chi_{184}(91,·)$, $\chi_{184}(165,·)$, $\chi_{184}(177,·)$, $\chi_{184}(41,·)$, $\chi_{184}(171,·)$, $\chi_{184}(173,·)$, $\chi_{184}(175,·)$, $\chi_{184}(49,·)$, $\chi_{184}(51,·)$, $\chi_{184}(183,·)$, $\chi_{184}(63,·)$, $\chi_{184}(67,·)$, $\chi_{184}(73,·)$, $\chi_{184}(79,·)$, $\chi_{184}(77,·)$, $\chi_{184}(141,·)$, $\chi_{184}(81,·)$, $\chi_{184}(83,·)$, $\chi_{184}(85,·)$, $\chi_{184}(143,·)$, $\chi_{184}(135,·)$, $\chi_{184}(93,·)$, $\chi_{184}(99,·)$, $\chi_{184}(101,·)$, $\chi_{184}(103,·)$, $\chi_{184}(105,·)$, $\chi_{184}(107,·)$, $\chi_{184}(111,·)$, $\chi_{184}(117,·)$, $\chi_{184}(169,·)$, $\chi_{184}(121,·)$$\rbrace$
|
| This is not a CM field. |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$
Trivial group, which has order $1$
(assuming GRH)
sage: K.class_group().invariants()
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | | $43$
|
|
| Torsion generator: | | \( -1 \) (order $2$)
| sage: UK.torsion_generator() 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() magma: [K!f(g): g in Generators(UK)]; |
| Regulator: | | \( 8626989567142591000000000000 \)
(assuming GRH)
|
|
$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{44}\cdot(2\pi)^{0}\cdot 8626989567142591000000000000 \cdot 1}{2\sqrt{482179487665033966874817964307376476160282778171317425736173928285756467369658548224}}\approx 0.109281015182506$ (assuming GRH)
$C_2\times C_{22}$ (as 44T2):
sage: K.galois_group(type='pari')
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
| $p$ |
$2$ |
$3$ |
$5$ |
$7$ |
$11$ |
$13$ |
$17$ |
$19$ |
$23$ |
$29$ |
$31$ |
$37$ |
$41$ |
$43$ |
$47$ |
$53$ |
$59$ |
|---|
| Cycle type |
R |
$22^{2}$ |
$22^{2}$ |
${\href{/LocalNumberField/7.11.0.1}{11} }^{4}$ |
$22^{2}$ |
$22^{2}$ |
$22^{2}$ |
$22^{2}$ |
R |
$22^{2}$ |
$22^{2}$ |
$22^{2}$ |
${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ |
$22^{2}$ |
${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ |
$22^{2}$ |
$22^{2}$ |
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];
