sage: x = polygen(QQ); K.<a> = NumberField(x^46 - 235*x^44 + 25850*x^42 - 1768375*x^40 + 84306250*x^38 - 2974806250*x^36 + 80537437500*x^34 - 1711420546875*x^32 + 28962501562500*x^30 - 393788398437500*x^28 + 4321029453125000*x^26 - 38300033789062500*x^24 + 273571669921875000*x^22 - 1565919287109375000*x^20 + 7117814941406250000*x^18 - 25357215728759765625*x^16 + 69527849578857421875*x^14 - 143145572662353515625*x^12 + 213895683288574218750*x^10 - 221132755279541015625*x^8 + 147421836853027343750*x^6 - 56700706481933593750*x^4 + 10309219360351562500*x^2 - 560283660888671875)
gp: K = bnfinit(y^46 - 235*y^44 + 25850*y^42 - 1768375*y^40 + 84306250*y^38 - 2974806250*y^36 + 80537437500*y^34 - 1711420546875*y^32 + 28962501562500*y^30 - 393788398437500*y^28 + 4321029453125000*y^26 - 38300033789062500*y^24 + 273571669921875000*y^22 - 1565919287109375000*y^20 + 7117814941406250000*y^18 - 25357215728759765625*y^16 + 69527849578857421875*y^14 - 143145572662353515625*y^12 + 213895683288574218750*y^10 - 221132755279541015625*y^8 + 147421836853027343750*y^6 - 56700706481933593750*y^4 + 10309219360351562500*y^2 - 560283660888671875, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^46 - 235*x^44 + 25850*x^42 - 1768375*x^40 + 84306250*x^38 - 2974806250*x^36 + 80537437500*x^34 - 1711420546875*x^32 + 28962501562500*x^30 - 393788398437500*x^28 + 4321029453125000*x^26 - 38300033789062500*x^24 + 273571669921875000*x^22 - 1565919287109375000*x^20 + 7117814941406250000*x^18 - 25357215728759765625*x^16 + 69527849578857421875*x^14 - 143145572662353515625*x^12 + 213895683288574218750*x^10 - 221132755279541015625*x^8 + 147421836853027343750*x^6 - 56700706481933593750*x^4 + 10309219360351562500*x^2 - 560283660888671875);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 - 235*x^44 + 25850*x^42 - 1768375*x^40 + 84306250*x^38 - 2974806250*x^36 + 80537437500*x^34 - 1711420546875*x^32 + 28962501562500*x^30 - 393788398437500*x^28 + 4321029453125000*x^26 - 38300033789062500*x^24 + 273571669921875000*x^22 - 1565919287109375000*x^20 + 7117814941406250000*x^18 - 25357215728759765625*x^16 + 69527849578857421875*x^14 - 143145572662353515625*x^12 + 213895683288574218750*x^10 - 221132755279541015625*x^8 + 147421836853027343750*x^6 - 56700706481933593750*x^4 + 10309219360351562500*x^2 - 560283660888671875)
\( x^{46} - 235 x^{44} + 25850 x^{42} - 1768375 x^{40} + 84306250 x^{38} - 2974806250 x^{36} + \cdots - 56\!\cdots\!75 \)
sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree: | | $46$ |
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Signature: | | $[46, 0]$ |
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Discriminant: | |
\(147\!\cdots\!000\)
\(\medspace = 2^{46}\cdot 5^{23}\cdot 47^{45}\)
|
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | | \(193.31\) | 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))
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Galois root discriminant: | | $2\cdot 5^{1/2}47^{45/46}\approx 193.31381977504472$
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Ramified primes: | |
\(2\), \(5\), \(47\)
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gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | | \(\Q(\sqrt{235}) \)
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$\card{ \Gal(K/\Q) }$: | | $46$ |
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This field is Galois and abelian over $\Q$. |
Conductor: | | \(940=2^{2}\cdot 5\cdot 47\) |
Dirichlet character group:
| |
$\lbrace$$\chi_{940}(1,·)$, $\chi_{940}(901,·)$, $\chi_{940}(521,·)$, $\chi_{940}(139,·)$, $\chi_{940}(399,·)$, $\chi_{940}(401,·)$, $\chi_{940}(19,·)$, $\chi_{940}(21,·)$, $\chi_{940}(279,·)$, $\chi_{940}(921,·)$, $\chi_{940}(539,·)$, $\chi_{940}(541,·)$, $\chi_{940}(801,·)$, $\chi_{940}(419,·)$, $\chi_{940}(39,·)$, $\chi_{940}(939,·)$, $\chi_{940}(179,·)$, $\chi_{940}(121,·)$, $\chi_{940}(441,·)$, $\chi_{940}(699,·)$, $\chi_{940}(61,·)$, $\chi_{940}(579,·)$, $\chi_{940}(581,·)$, $\chi_{940}(839,·)$, $\chi_{940}(841,·)$, $\chi_{940}(859,·)$, $\chi_{940}(721,·)$, $\chi_{940}(339,·)$, $\chi_{940}(919,·)$, $\chi_{940}(341,·)$, $\chi_{940}(599,·)$, $\chi_{940}(601,·)$, $\chi_{940}(219,·)$, $\chi_{940}(199,·)$, $\chi_{940}(101,·)$, $\chi_{940}(99,·)$, $\chi_{940}(741,·)$, $\chi_{940}(81,·)$, $\chi_{940}(361,·)$, $\chi_{940}(359,·)$, $\chi_{940}(879,·)$, $\chi_{940}(241,·)$, $\chi_{940}(499,·)$, $\chi_{940}(761,·)$, $\chi_{940}(819,·)$, $\chi_{940}(661,·)$$\rbrace$
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This is not a CM field. |
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{25}a^{4}$, $\frac{1}{25}a^{5}$, $\frac{1}{125}a^{6}$, $\frac{1}{125}a^{7}$, $\frac{1}{625}a^{8}$, $\frac{1}{625}a^{9}$, $\frac{1}{3125}a^{10}$, $\frac{1}{3125}a^{11}$, $\frac{1}{15625}a^{12}$, $\frac{1}{15625}a^{13}$, $\frac{1}{78125}a^{14}$, $\frac{1}{78125}a^{15}$, $\frac{1}{390625}a^{16}$, $\frac{1}{390625}a^{17}$, $\frac{1}{1953125}a^{18}$, $\frac{1}{1953125}a^{19}$, $\frac{1}{9765625}a^{20}$, $\frac{1}{9765625}a^{21}$, $\frac{1}{48828125}a^{22}$, $\frac{1}{48828125}a^{23}$, $\frac{1}{244140625}a^{24}$, $\frac{1}{244140625}a^{25}$, $\frac{1}{1220703125}a^{26}$, $\frac{1}{1220703125}a^{27}$, $\frac{1}{6103515625}a^{28}$, $\frac{1}{6103515625}a^{29}$, $\frac{1}{30517578125}a^{30}$, $\frac{1}{30517578125}a^{31}$, $\frac{1}{152587890625}a^{32}$, $\frac{1}{152587890625}a^{33}$, $\frac{1}{762939453125}a^{34}$, $\frac{1}{762939453125}a^{35}$, $\frac{1}{3814697265625}a^{36}$, $\frac{1}{3814697265625}a^{37}$, $\frac{1}{19073486328125}a^{38}$, $\frac{1}{19073486328125}a^{39}$, $\frac{1}{95367431640625}a^{40}$, $\frac{1}{95367431640625}a^{41}$, $\frac{1}{476837158203125}a^{42}$, $\frac{1}{476837158203125}a^{43}$, $\frac{1}{23\!\cdots\!25}a^{44}$, $\frac{1}{23\!\cdots\!25}a^{45}$
not computed
sage: K.class_group().invariants()
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank: | | $45$
|
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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)
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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 - 235*x^44 + 25850*x^42 - 1768375*x^40 + 84306250*x^38 - 2974806250*x^36 + 80537437500*x^34 - 1711420546875*x^32 + 28962501562500*x^30 - 393788398437500*x^28 + 4321029453125000*x^26 - 38300033789062500*x^24 + 273571669921875000*x^22 - 1565919287109375000*x^20 + 7117814941406250000*x^18 - 25357215728759765625*x^16 + 69527849578857421875*x^14 - 143145572662353515625*x^12 + 213895683288574218750*x^10 - 221132755279541015625*x^8 + 147421836853027343750*x^6 - 56700706481933593750*x^4 + 10309219360351562500*x^2 - 560283660888671875) 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 - 235*x^44 + 25850*x^42 - 1768375*x^40 + 84306250*x^38 - 2974806250*x^36 + 80537437500*x^34 - 1711420546875*x^32 + 28962501562500*x^30 - 393788398437500*x^28 + 4321029453125000*x^26 - 38300033789062500*x^24 + 273571669921875000*x^22 - 1565919287109375000*x^20 + 7117814941406250000*x^18 - 25357215728759765625*x^16 + 69527849578857421875*x^14 - 143145572662353515625*x^12 + 213895683288574218750*x^10 - 221132755279541015625*x^8 + 147421836853027343750*x^6 - 56700706481933593750*x^4 + 10309219360351562500*x^2 - 560283660888671875, 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 - 235*x^44 + 25850*x^42 - 1768375*x^40 + 84306250*x^38 - 2974806250*x^36 + 80537437500*x^34 - 1711420546875*x^32 + 28962501562500*x^30 - 393788398437500*x^28 + 4321029453125000*x^26 - 38300033789062500*x^24 + 273571669921875000*x^22 - 1565919287109375000*x^20 + 7117814941406250000*x^18 - 25357215728759765625*x^16 + 69527849578857421875*x^14 - 143145572662353515625*x^12 + 213895683288574218750*x^10 - 221132755279541015625*x^8 + 147421836853027343750*x^6 - 56700706481933593750*x^4 + 10309219360351562500*x^2 - 560283660888671875); 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 - 235*x^44 + 25850*x^42 - 1768375*x^40 + 84306250*x^38 - 2974806250*x^36 + 80537437500*x^34 - 1711420546875*x^32 + 28962501562500*x^30 - 393788398437500*x^28 + 4321029453125000*x^26 - 38300033789062500*x^24 + 273571669921875000*x^22 - 1565919287109375000*x^20 + 7117814941406250000*x^18 - 25357215728759765625*x^16 + 69527849578857421875*x^14 - 143145572662353515625*x^12 + 213895683288574218750*x^10 - 221132755279541015625*x^8 + 147421836853027343750*x^6 - 56700706481933593750*x^4 + 10309219360351562500*x^2 - 560283660888671875); 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 |
$23^{2}$ |
R |
$23^{2}$ |
$23^{2}$ |
$23^{2}$ |
$46$ |
$23^{2}$ |
$46$ |
$46$ |
$23^{2}$ |
$46$ |
$46$ |
$46$ |
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]
|