sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 42*x^40 + 819*x^38 - 9842*x^36 + 81585*x^34 - 494802*x^32 + 2272424*x^30 - 8069425*x^28 + 22428280*x^26 - 49085750*x^24 + 84674891*x^22 - 114729727*x^20 + 121131479*x^18 - 98380632*x^16 + 60329941*x^14 - 27217932*x^12 + 8716708*x^10 - 1885324*x^8 + 256221*x^6 - 19551*x^4 + 686*x^2 - 7)
gp: K = bnfinit(x^42 - 42*x^40 + 819*x^38 - 9842*x^36 + 81585*x^34 - 494802*x^32 + 2272424*x^30 - 8069425*x^28 + 22428280*x^26 - 49085750*x^24 + 84674891*x^22 - 114729727*x^20 + 121131479*x^18 - 98380632*x^16 + 60329941*x^14 - 27217932*x^12 + 8716708*x^10 - 1885324*x^8 + 256221*x^6 - 19551*x^4 + 686*x^2 - 7, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 686, 0, -19551, 0, 256221, 0, -1885324, 0, 8716708, 0, -27217932, 0, 60329941, 0, -98380632, 0, 121131479, 0, -114729727, 0, 84674891, 0, -49085750, 0, 22428280, 0, -8069425, 0, 2272424, 0, -494802, 0, 81585, 0, -9842, 0, 819, 0, -42, 0, 1]);
\( x^{42} - 42 x^{40} + 819 x^{38} - 9842 x^{36} + 81585 x^{34} - 494802 x^{32} + 2272424 x^{30} - 8069425 x^{28} + 22428280 x^{26} - 49085750 x^{24} + 84674891 x^{22} - 114729727 x^{20} + 121131479 x^{18} - 98380632 x^{16} + 60329941 x^{14} - 27217932 x^{12} + 8716708 x^{10} - 1885324 x^{8} + 256221 x^{6} - 19551 x^{4} + 686 x^{2} - 7 \)
sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
| Degree: | | $42$ |
|
| Signature: | | $[42, 0]$ |
|
| Discriminant: | | \(519\!\cdots\!528\)\(\medspace = 2^{42}\cdot 7^{77}\) |
magma: Discriminant(Integers(K)); |
| Root discriminant: | | $70.86$ | 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: | | $2, 7$ |
gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(Integers(K))); |
| $|\Gal(K/\Q)|$: | | $42$ |
| This field is Galois and abelian over $\Q$. |
| Conductor: | | \(196=2^{2}\cdot 7^{2}\) |
| Dirichlet character group:
| |
$\lbrace$$\chi_{196}(1,·)$, $\chi_{196}(3,·)$, $\chi_{196}(65,·)$, $\chi_{196}(9,·)$, $\chi_{196}(139,·)$, $\chi_{196}(141,·)$, $\chi_{196}(143,·)$, $\chi_{196}(19,·)$, $\chi_{196}(149,·)$, $\chi_{196}(25,·)$, $\chi_{196}(27,·)$, $\chi_{196}(29,·)$, $\chi_{196}(31,·)$, $\chi_{196}(37,·)$, $\chi_{196}(167,·)$, $\chi_{196}(169,·)$, $\chi_{196}(171,·)$, $\chi_{196}(47,·)$, $\chi_{196}(177,·)$, $\chi_{196}(53,·)$, $\chi_{196}(137,·)$, $\chi_{196}(57,·)$, $\chi_{196}(159,·)$, $\chi_{196}(193,·)$, $\chi_{196}(195,·)$, $\chi_{196}(55,·)$, $\chi_{196}(81,·)$, $\chi_{196}(75,·)$, $\chi_{196}(83,·)$, $\chi_{196}(85,·)$, $\chi_{196}(87,·)$, $\chi_{196}(93,·)$, $\chi_{196}(165,·)$, $\chi_{196}(59,·)$, $\chi_{196}(131,·)$, $\chi_{196}(103,·)$, $\chi_{196}(109,·)$, $\chi_{196}(111,·)$, $\chi_{196}(113,·)$, $\chi_{196}(115,·)$, $\chi_{196}(187,·)$, $\chi_{196}(121,·)$$\rbrace$
|
| This is not a CM field. |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$
Trivial group, which has order $1$
(assuming GRH)
sage: K.class_group().invariants()
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | | $41$
|
|
| 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: | | \( 40972406918644810000000000 \)
(assuming GRH)
|
|
$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{42}\cdot(2\pi)^{0}\cdot 40972406918644810000000000 \cdot 1}{2\sqrt{519767234928222794437622861788597020192717533652199079454480438860528408854528}}\approx 0.124973188408209$ (assuming GRH)
$C_{42}$ (as 42T1):
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 |
$21^{2}$ |
$42$ |
R |
$42$ |
${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ |
$42$ |
${\href{/LocalNumberField/19.3.0.1}{3} }^{14}$ |
$42$ |
${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$ |
${\href{/LocalNumberField/31.3.0.1}{3} }^{14}$ |
$21^{2}$ |
${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ |
${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ |
$21^{2}$ |
$21^{2}$ |
$21^{2}$ |
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];
