sage: x = polygen(QQ); K.<a> = NumberField(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576)
gp: K = bnfinit(y^46 + 94*y^44 + 4136*y^42 + 113176*y^40 + 2158240*y^38 + 30462016*y^36 + 329881344*y^34 + 2803991424*y^32 + 18980865024*y^30 + 103229265920*y^28 + 453092777984*y^26 + 1606419849216*y^24 + 4589770997760*y^22 + 10508706447360*y^20 + 19106738995200*y^18 + 27227103068160*y^16 + 29861984010240*y^14 + 24592222126080*y^12 + 14698799431680*y^10 + 6078450892800*y^8 + 1620920238080*y^6 + 249372344320*y^4 + 18136170496*y^2 + 394264576, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576)
\( x^{46} + 94 x^{44} + 4136 x^{42} + 113176 x^{40} + 2158240 x^{38} + 30462016 x^{36} + 329881344 x^{34} + \cdots + 394264576 \)
sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree: | | $46$ |
|
Signature: | | $[0, 23]$ |
|
Discriminant: | |
\(-103\!\cdots\!184\)
\(\medspace = -\,2^{69}\cdot 47^{45}\)
|
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
|
Root discriminant: | | \(122.26\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
|
Galois root discriminant: | | $2^{3/2}47^{45/46}\approx 122.26239473528804$
|
Ramified primes: | |
\(2\), \(47\)
|
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
|
Discriminant root field: | | \(\Q(\sqrt{-94}) \)
|
$\card{ \Gal(K/\Q) }$: | | $46$ |
|
This field is Galois and abelian over $\Q$. |
Conductor: | | \(376=2^{3}\cdot 47\) |
Dirichlet character group:
| |
$\lbrace$$\chi_{376}(1,·)$, $\chi_{376}(5,·)$, $\chi_{376}(81,·)$, $\chi_{376}(145,·)$, $\chi_{376}(9,·)$, $\chi_{376}(13,·)$, $\chi_{376}(45,·)$, $\chi_{376}(17,·)$, $\chi_{376}(89,·)$, $\chi_{376}(25,·)$, $\chi_{376}(29,·)$, $\chi_{376}(69,·)$, $\chi_{376}(261,·)$, $\chi_{376}(133,·)$, $\chi_{376}(289,·)$, $\chi_{376}(293,·)$, $\chi_{376}(49,·)$, $\chi_{376}(169,·)$, $\chi_{376}(301,·)$, $\chi_{376}(93,·)$, $\chi_{376}(177,·)$, $\chi_{376}(181,·)$, $\chi_{376}(117,·)$, $\chi_{376}(317,·)$, $\chi_{376}(245,·)$, $\chi_{376}(65,·)$, $\chi_{376}(325,·)$, $\chi_{376}(97,·)$, $\chi_{376}(77,·)$, $\chi_{376}(209,·)$, $\chi_{376}(85,·)$, $\chi_{376}(121,·)$, $\chi_{376}(345,·)$, $\chi_{376}(349,·)$, $\chi_{376}(353,·)$, $\chi_{376}(229,·)$, $\chi_{376}(337,·)$, $\chi_{376}(361,·)$, $\chi_{376}(225,·)$, $\chi_{376}(109,·)$, $\chi_{376}(241,·)$, $\chi_{376}(221,·)$, $\chi_{376}(373,·)$, $\chi_{376}(153,·)$, $\chi_{376}(249,·)$, $\chi_{376}(125,·)$$\rbrace$
|
This is a CM field. |
Reflex fields: | | unavailable$^{4194304}$ |
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$, $\frac{1}{262144}a^{36}$, $\frac{1}{262144}a^{37}$, $\frac{1}{524288}a^{38}$, $\frac{1}{524288}a^{39}$, $\frac{1}{1048576}a^{40}$, $\frac{1}{1048576}a^{41}$, $\frac{1}{2097152}a^{42}$, $\frac{1}{2097152}a^{43}$, $\frac{1}{4194304}a^{44}$, $\frac{1}{4194304}a^{45}$
not computed
sage: K.class_group().invariants()
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank: | | $22$
|
|
Torsion generator: | |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
|
Fundamental units: | | not computed
| sage: UK.fundamental_units()
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
|
Regulator: | | not computed
|
|
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
not computed
\end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
$C_{46}$ (as 46T1):
sage: K.galois_group(type='pari')
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
$p$ |
$2$ |
$3$ |
$5$ |
$7$ |
$11$ |
$13$ |
$17$ |
$19$ |
$23$ |
$29$ |
$31$ |
$37$ |
$41$ |
$43$ |
$47$ |
$53$ |
$59$ |
Cycle type |
R |
$46$ |
$23^{2}$ |
$23^{2}$ |
$23^{2}$ |
$23^{2}$ |
$23^{2}$ |
$23^{2}$ |
$46$ |
$23^{2}$ |
$46$ |
$46$ |
$46$ |
$23^{2}$ |
R |
$46$ |
$46$ |
In the table, R denotes a ramified prime.
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
|