Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^37 + 2*x - 2)
gp: K = bnfinit(y^37 + 2*y - 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^37 + 2*x - 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^37 + 2*x - 2)
\( x^{37} + 2x - 2 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $37$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(739965118312037415559767325366968776023307778410798181434074756808704\)
\(\medspace = 2^{36}\cdot 11\cdot 13\cdot 29\cdot 8111\cdot 32\!\cdots\!17\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(72.67\) | 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: | $2^{36/37}11^{1/2}13^{1/2}29^{1/2}8111^{1/2}320127488203456295874894895225917443149549690432217^{1/2}\approx 2.0368533035272427e+29$ | ||
| Ramified primes: |
\(2\), \(11\), \(13\), \(29\), \(8111\), \(32012\!\cdots\!32217\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{10767\!\cdots\!24789}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$a-1$, $a^{25}-a^{13}+a-1$, $a^{28}+a^{19}-a+1$, $a^{36}+a^{34}+a^{32}+a^{30}+a^{28}+a^{26}+a^{24}+a^{22}+a^{20}+a^{18}+a^{16}+a^{14}+a^{12}+a^{10}+a^{8}+a^{6}+a^{4}+a+1$, $3a^{36}+2a^{35}+3a^{34}+2a^{33}+2a^{32}+2a^{31}+a^{30}+2a^{29}+2a^{27}+2a^{25}+a^{24}+a^{23}+a^{22}+a^{20}+a^{18}+a^{17}+2a^{15}-a^{14}+2a^{13}-a^{12}+2a^{11}+a^{9}+a^{8}-a^{7}+2a^{6}-a^{5}+2a^{4}+a^{2}+a+5$, $2a^{36}+3a^{35}+2a^{34}+2a^{33}+3a^{32}+a^{31}+2a^{30}+2a^{29}+2a^{27}+a^{26}+a^{25}+2a^{24}-a^{23}+a^{22}+a^{21}-a^{20}+2a^{19}+a^{16}-2a^{15}+a^{14}+a^{13}-a^{12}+2a^{11}-a^{10}-a^{9}+2a^{8}-2a^{7}+2a^{6}+a^{5}-2a^{4}+3a^{3}-2a^{2}+7$, $2a^{36}+3a^{35}+a^{34}+2a^{33}+3a^{32}+a^{31}+2a^{30}+2a^{29}+2a^{27}+2a^{26}+2a^{24}+a^{23}+2a^{21}+3a^{18}+2a^{15}-a^{14}+a^{13}+3a^{12}-2a^{11}+2a^{9}-a^{8}+2a^{7}+a^{6}-3a^{5}+2a^{4}+a^{3}-a^{2}+3a+3$, $a^{36}+3a^{35}-a^{34}-a^{33}+a^{32}+2a^{31}-2a^{30}-a^{29}+3a^{28}+a^{27}-2a^{26}-a^{25}+3a^{24}-3a^{22}+2a^{21}+3a^{20}-4a^{18}+3a^{17}+2a^{16}-3a^{15}-2a^{14}+4a^{13}+3a^{12}-5a^{11}+4a^{9}-a^{8}-5a^{7}+a^{6}+7a^{5}-4a^{4}-3a^{3}+3a^{2}+4a-3$, $2a^{36}+3a^{35}+2a^{34}+a^{33}-2a^{31}-2a^{30}-4a^{29}-2a^{28}-3a^{27}-a^{26}+a^{25}+a^{24}+3a^{23}+2a^{22}+4a^{21}+a^{19}-a^{18}-3a^{17}-2a^{16}-4a^{15}-a^{14}-3a^{13}+2a^{12}+3a^{10}+3a^{9}+2a^{8}+2a^{7}+a^{6}-a^{5}-2a^{4}-3a^{3}-a^{2}-5a+5$, $6a^{36}+5a^{35}+5a^{34}+6a^{33}+7a^{32}+7a^{31}+6a^{30}+5a^{29}+4a^{28}+4a^{27}+5a^{26}+5a^{25}+5a^{24}+5a^{23}+4a^{22}+4a^{21}+4a^{20}+3a^{19}+2a^{18}+a^{17}-a^{13}-3a^{12}-5a^{11}-6a^{10}-6a^{9}-4a^{8}-2a^{7}-3a^{6}-4a^{5}-5a^{4}-6a^{3}-3a^{2}+13$, $2a^{36}+2a^{35}+3a^{34}+3a^{33}+3a^{32}+3a^{31}+2a^{30}+3a^{29}+a^{28}+a^{27}+a^{26}-2a^{25}-a^{24}-3a^{23}-4a^{22}-3a^{21}-4a^{20}-2a^{19}-2a^{18}+a^{16}+3a^{14}+a^{13}+2a^{12}+3a^{11}+a^{10}+3a^{9}+a^{8}+2a^{7}+a^{6}-a^{5}+a^{4}-4a^{3}-2a^{2}-2a-1$, $a^{36}+2a^{35}-a^{34}-2a^{33}-a^{32}+2a^{31}+a^{30}-a^{29}-3a^{28}+a^{27}+2a^{26}+a^{25}-3a^{24}-2a^{23}+a^{22}+3a^{21}-a^{20}-3a^{19}-2a^{18}+3a^{17}+2a^{16}-a^{15}-4a^{14}+3a^{12}+2a^{11}-3a^{10}-3a^{9}+2a^{8}+4a^{7}+a^{6}-5a^{5}-2a^{4}+3a^{3}+4a^{2}-2a-3$, $4a^{36}+4a^{35}+4a^{34}+3a^{33}+4a^{32}+5a^{31}+3a^{30}+2a^{29}+2a^{28}+a^{27}+a^{26}+2a^{25}+a^{24}-a^{22}-2a^{21}-a^{20}-a^{18}-a^{17}-2a^{16}-3a^{15}-3a^{14}-2a^{13}-a^{11}-3a^{10}-a^{9}-2a^{8}-3a^{7}+a^{6}-2a^{4}-2a^{2}-a+11$, $a^{36}-2a^{33}+a^{31}-a^{29}+2a^{27}-a^{25}-a^{24}+3a^{23}+a^{22}-a^{20}+3a^{19}+a^{18}-2a^{17}+a^{15}+3a^{14}-a^{12}+4a^{10}-3a^{9}-a^{8}+3a^{6}+2a^{5}-4a^{4}+3a^{2}-3$, $a^{36}+2a^{35}+3a^{34}+2a^{33}-2a^{32}-3a^{31}-2a^{30}+3a^{28}+4a^{27}+a^{26}-a^{25}-a^{24}-2a^{23}-a^{22}+a^{21}+a^{20}+a^{19}+3a^{18}-3a^{16}-3a^{15}-2a^{14}+5a^{12}+3a^{11}-a^{10}-3a^{9}-3a^{8}-3a^{7}+2a^{6}+3a^{5}+a^{4}+a^{3}+2a^{2}-3a-1$, $18a^{36}+11a^{35}+9a^{34}+14a^{33}+22a^{32}+26a^{31}+24a^{30}+17a^{29}+7a^{28}+a^{27}+3a^{26}+10a^{25}+18a^{24}+22a^{23}+18a^{22}+8a^{21}-a^{20}-4a^{19}-a^{18}+9a^{17}+18a^{16}+18a^{15}+13a^{14}+3a^{13}-9a^{12}-8a^{11}+a^{10}+9a^{9}+17a^{8}+18a^{7}+7a^{6}-4a^{5}-11a^{4}-11a^{3}+a^{2}+14a+53$, $2a^{36}+2a^{34}-2a^{33}-2a^{32}-2a^{30}-5a^{28}-3a^{27}-a^{26}-3a^{25}-5a^{23}-2a^{22}-a^{21}-3a^{20}+a^{19}-4a^{18}+a^{16}-2a^{15}+2a^{14}-4a^{13}+3a^{12}+3a^{11}-a^{10}+3a^{9}-4a^{8}+5a^{7}+3a^{6}+4a^{4}-4a^{3}+6a^{2}+2a+3$, $7a^{36}+3a^{35}+2a^{34}+4a^{33}+4a^{32}+2a^{31}+8a^{30}+5a^{29}+4a^{28}+5a^{27}+4a^{26}-2a^{25}+2a^{24}+3a^{23}-2a^{22}+4a^{21}+6a^{20}+3a^{19}+a^{18}+7a^{17}-2a^{16}-a^{15}+a^{14}-a^{12}+4a^{11}+4a^{10}+a^{9}+6a^{8}-2a^{7}+a^{6}-a^{5}-a^{4}-5a^{3}+7a^{2}+13$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 109736008063969670000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{18}\cdot 109736008063969670000 \cdot 1}{2\cdot\sqrt{739965118312037415559767325366968776023307778410798181434074756808704}}\cr\approx \mathstrut & 0.939682554515302 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^37 + 2*x - 2)
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^37 + 2*x - 2, 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^37 + 2*x - 2);
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^37 + 2*x - 2);
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))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 13763753091226345046315979581580902400000000 |
| The 21637 conjugacy class representatives for $S_{37}$ are not computed |
| Character table for $S_{37}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $29{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $19{,}\,17{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | R | $32{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.12.0.1}{12} }^{2}{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | R | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | $36{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $37$ | $24{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $37$ | $37$ | $1$ | $36$ | |||
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.7.0.1 | $x^{7} + 4 x + 9$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 11.7.0.1 | $x^{7} + 4 x + 9$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 11.17.0.1 | $x^{17} + 4 x + 9$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
|
\(13\)
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.8.0.1 | $x^{8} + 8 x^{4} + 12 x^{3} + 2 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 13.16.0.1 | $x^{16} + 3 x^{8} + 12 x^{7} + 8 x^{6} + 2 x^{5} + 12 x^{4} + 9 x^{3} + 12 x^{2} + 6 x + 2$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.9.0.1 | $x^{9} + 4 x^{3} + 22 x^{2} + 22 x + 27$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 29.10.0.1 | $x^{10} + x^{6} + 25 x^{5} + 8 x^{4} + 17 x^{3} + 2 x^{2} + 22 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 29.13.0.1 | $x^{13} + 7 x + 27$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
|
\(8111\)
| $\Q_{8111}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{8111}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
|
\(320\!\cdots\!217\)
| $\Q_{32\!\cdots\!17}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
| Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |