↑

Properties

Label 46.0.102...983.1
Degree $46$
Signature $[0, 23]$
Discriminant $-1.022\times 10^{93}$
Root discriminant \(105.18\)
Ramified primes $7,47$
Class number not computed
Class group not computed
Galois group $C_{46}$ (as 46T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 + 68*x^44 - 61*x^43 + 2272*x^42 - 1838*x^41 + 49088*x^40 - 35837*x^39 + 764082*x^38 - 502716*x^37 + 9061304*x^36 - 5356876*x^35 + 84584992*x^34 - 44739144*x^33 + 634011776*x^32 - 298356295*x^31 + 3862036333*x^30 - 1605788227*x^29 + 19242860452*x^28 - 7010057107*x^27 + 78601898736*x^26 - 24835843890*x^25 + 262854128576*x^24 - 71169036677*x^23 + 716241923902*x^22 - 163773410984*x^21 + 1577302832552*x^20 - 299245739680*x^19 + 2773773179200*x^18 - 427289762048*x^17 + 3831228318848*x^16 - 466369885184*x^15 + 4064622286848*x^14 - 377834643456*x^13 + 3213707960320*x^12 - 217643319296*x^11 + 1816862654464*x^10 - 84719697920*x^9 + 692213350400*x^8 - 19502399488*x^7 + 162748039168*x^6 - 3073900544*x^5 + 20306198528*x^4 + 161480704*x^3 + 1019215872*x^2 - 50331648*x + 8388608)
 
gp: K = bnfinit(y^46 - y^45 + 68*y^44 - 61*y^43 + 2272*y^42 - 1838*y^41 + 49088*y^40 - 35837*y^39 + 764082*y^38 - 502716*y^37 + 9061304*y^36 - 5356876*y^35 + 84584992*y^34 - 44739144*y^33 + 634011776*y^32 - 298356295*y^31 + 3862036333*y^30 - 1605788227*y^29 + 19242860452*y^28 - 7010057107*y^27 + 78601898736*y^26 - 24835843890*y^25 + 262854128576*y^24 - 71169036677*y^23 + 716241923902*y^22 - 163773410984*y^21 + 1577302832552*y^20 - 299245739680*y^19 + 2773773179200*y^18 - 427289762048*y^17 + 3831228318848*y^16 - 466369885184*y^15 + 4064622286848*y^14 - 377834643456*y^13 + 3213707960320*y^12 - 217643319296*y^11 + 1816862654464*y^10 - 84719697920*y^9 + 692213350400*y^8 - 19502399488*y^7 + 162748039168*y^6 - 3073900544*y^5 + 20306198528*y^4 + 161480704*y^3 + 1019215872*y^2 - 50331648*y + 8388608, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^46 - x^45 + 68*x^44 - 61*x^43 + 2272*x^42 - 1838*x^41 + 49088*x^40 - 35837*x^39 + 764082*x^38 - 502716*x^37 + 9061304*x^36 - 5356876*x^35 + 84584992*x^34 - 44739144*x^33 + 634011776*x^32 - 298356295*x^31 + 3862036333*x^30 - 1605788227*x^29 + 19242860452*x^28 - 7010057107*x^27 + 78601898736*x^26 - 24835843890*x^25 + 262854128576*x^24 - 71169036677*x^23 + 716241923902*x^22 - 163773410984*x^21 + 1577302832552*x^20 - 299245739680*x^19 + 2773773179200*x^18 - 427289762048*x^17 + 3831228318848*x^16 - 466369885184*x^15 + 4064622286848*x^14 - 377834643456*x^13 + 3213707960320*x^12 - 217643319296*x^11 + 1816862654464*x^10 - 84719697920*x^9 + 692213350400*x^8 - 19502399488*x^7 + 162748039168*x^6 - 3073900544*x^5 + 20306198528*x^4 + 161480704*x^3 + 1019215872*x^2 - 50331648*x + 8388608);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 - x^45 + 68*x^44 - 61*x^43 + 2272*x^42 - 1838*x^41 + 49088*x^40 - 35837*x^39 + 764082*x^38 - 502716*x^37 + 9061304*x^36 - 5356876*x^35 + 84584992*x^34 - 44739144*x^33 + 634011776*x^32 - 298356295*x^31 + 3862036333*x^30 - 1605788227*x^29 + 19242860452*x^28 - 7010057107*x^27 + 78601898736*x^26 - 24835843890*x^25 + 262854128576*x^24 - 71169036677*x^23 + 716241923902*x^22 - 163773410984*x^21 + 1577302832552*x^20 - 299245739680*x^19 + 2773773179200*x^18 - 427289762048*x^17 + 3831228318848*x^16 - 466369885184*x^15 + 4064622286848*x^14 - 377834643456*x^13 + 3213707960320*x^12 - 217643319296*x^11 + 1816862654464*x^10 - 84719697920*x^9 + 692213350400*x^8 - 19502399488*x^7 + 162748039168*x^6 - 3073900544*x^5 + 20306198528*x^4 + 161480704*x^3 + 1019215872*x^2 - 50331648*x + 8388608)
 

\( x^{46} - x^{45} + 68 x^{44} - 61 x^{43} + 2272 x^{42} - 1838 x^{41} + 49088 x^{40} - 35837 x^{39} + \cdots + 8388608 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $46$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 23]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-102\!\cdots\!983\) \(\medspace = -\,7^{23}\cdot 47^{44}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(105.18\)
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:  $7^{1/2}47^{22/23}\approx 105.1833433322819$
Ramified primes:   \(7\), \(47\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-7}) \)
$\card{ \Gal(K/\Q) }$:  $46$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(329=7\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{329}(1,·)$, $\chi_{329}(260,·)$, $\chi_{329}(6,·)$, $\chi_{329}(8,·)$, $\chi_{329}(267,·)$, $\chi_{329}(272,·)$, $\chi_{329}(148,·)$, $\chi_{329}(153,·)$, $\chi_{329}(27,·)$, $\chi_{329}(286,·)$, $\chi_{329}(288,·)$, $\chi_{329}(34,·)$, $\chi_{329}(155,·)$, $\chi_{329}(36,·)$, $\chi_{329}(169,·)$, $\chi_{329}(300,·)$, $\chi_{329}(48,·)$, $\chi_{329}(50,·)$, $\chi_{329}(307,·)$, $\chi_{329}(309,·)$, $\chi_{329}(183,·)$, $\chi_{329}(111,·)$, $\chi_{329}(314,·)$, $\chi_{329}(316,·)$, $\chi_{329}(190,·)$, $\chi_{329}(64,·)$, $\chi_{329}(195,·)$, $\chi_{329}(197,·)$, $\chi_{329}(71,·)$, $\chi_{329}(202,·)$, $\chi_{329}(55,·)$, $\chi_{329}(204,·)$, $\chi_{329}(162,·)$, $\chi_{329}(209,·)$, $\chi_{329}(83,·)$, $\chi_{329}(216,·)$, $\chi_{329}(97,·)$, $\chi_{329}(230,·)$, $\chi_{329}(106,·)$, $\chi_{329}(225,·)$, $\chi_{329}(237,·)$, $\chi_{329}(239,·)$, $\chi_{329}(244,·)$, $\chi_{329}(118,·)$, $\chi_{329}(251,·)$, $\chi_{329}(253,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{4194304}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{25}-\frac{1}{4}a^{24}-\frac{1}{4}a^{22}-\frac{1}{2}a^{20}-\frac{1}{4}a^{18}-\frac{1}{2}a^{17}+\frac{1}{4}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{26}-\frac{1}{8}a^{25}-\frac{1}{8}a^{23}-\frac{1}{4}a^{21}+\frac{3}{8}a^{19}+\frac{1}{4}a^{18}-\frac{1}{2}a^{15}+\frac{1}{8}a^{11}-\frac{3}{8}a^{10}+\frac{1}{8}a^{9}+\frac{1}{8}a^{7}+\frac{1}{4}a^{5}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{27}-\frac{1}{16}a^{26}-\frac{1}{16}a^{24}-\frac{1}{2}a^{23}+\frac{3}{8}a^{22}-\frac{1}{2}a^{21}-\frac{5}{16}a^{20}+\frac{1}{8}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{4}a^{16}-\frac{1}{2}a^{14}-\frac{7}{16}a^{12}-\frac{3}{16}a^{11}-\frac{7}{16}a^{10}-\frac{1}{2}a^{9}+\frac{1}{16}a^{8}-\frac{1}{2}a^{7}-\frac{3}{8}a^{6}-\frac{1}{2}a^{5}+\frac{3}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{28}-\frac{1}{32}a^{27}-\frac{1}{32}a^{25}-\frac{1}{4}a^{24}+\frac{3}{16}a^{23}+\frac{1}{4}a^{22}-\frac{5}{32}a^{21}+\frac{1}{16}a^{20}-\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{3}{8}a^{17}+\frac{1}{4}a^{15}-\frac{7}{32}a^{13}+\frac{13}{32}a^{12}-\frac{7}{32}a^{11}+\frac{1}{4}a^{10}-\frac{15}{32}a^{9}-\frac{1}{4}a^{8}-\frac{3}{16}a^{7}-\frac{1}{4}a^{6}+\frac{3}{32}a^{5}+\frac{7}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{29}-\frac{1}{64}a^{28}-\frac{1}{64}a^{26}-\frac{1}{8}a^{25}+\frac{3}{32}a^{24}-\frac{3}{8}a^{23}+\frac{27}{64}a^{22}-\frac{15}{32}a^{21}-\frac{1}{8}a^{20}+\frac{3}{8}a^{19}-\frac{3}{16}a^{18}-\frac{1}{2}a^{17}-\frac{3}{8}a^{16}-\frac{1}{2}a^{15}+\frac{25}{64}a^{14}-\frac{19}{64}a^{13}+\frac{25}{64}a^{12}-\frac{3}{8}a^{11}-\frac{15}{64}a^{10}+\frac{3}{8}a^{9}+\frac{13}{32}a^{8}+\frac{3}{8}a^{7}-\frac{29}{64}a^{6}+\frac{7}{32}a^{5}-\frac{5}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{30}-\frac{1}{128}a^{29}-\frac{1}{128}a^{27}-\frac{1}{16}a^{26}+\frac{3}{64}a^{25}-\frac{3}{16}a^{24}+\frac{27}{128}a^{23}-\frac{15}{64}a^{22}+\frac{7}{16}a^{21}-\frac{5}{16}a^{20}-\frac{3}{32}a^{19}+\frac{1}{4}a^{18}+\frac{5}{16}a^{17}-\frac{1}{4}a^{16}-\frac{39}{128}a^{15}+\frac{45}{128}a^{14}+\frac{25}{128}a^{13}-\frac{3}{16}a^{12}-\frac{15}{128}a^{11}-\frac{5}{16}a^{10}+\frac{13}{64}a^{9}-\frac{5}{16}a^{8}-\frac{29}{128}a^{7}+\frac{7}{64}a^{6}-\frac{5}{32}a^{5}-\frac{5}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{31}-\frac{1}{256}a^{30}-\frac{1}{256}a^{28}-\frac{1}{32}a^{27}+\frac{3}{128}a^{26}-\frac{3}{32}a^{25}+\frac{27}{256}a^{24}-\frac{15}{128}a^{23}+\frac{7}{32}a^{22}-\frac{5}{32}a^{21}+\frac{29}{64}a^{20}+\frac{1}{8}a^{19}-\frac{11}{32}a^{18}+\frac{3}{8}a^{17}+\frac{89}{256}a^{16}-\frac{83}{256}a^{15}-\frac{103}{256}a^{14}-\frac{3}{32}a^{13}-\frac{15}{256}a^{12}+\frac{11}{32}a^{11}+\frac{13}{128}a^{10}-\frac{5}{32}a^{9}+\frac{99}{256}a^{8}+\frac{7}{128}a^{7}-\frac{5}{64}a^{6}-\frac{5}{32}a^{5}+\frac{7}{16}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{512}a^{32}-\frac{1}{512}a^{31}-\frac{1}{512}a^{29}-\frac{1}{64}a^{28}+\frac{3}{256}a^{27}-\frac{3}{64}a^{26}+\frac{27}{512}a^{25}-\frac{15}{256}a^{24}+\frac{7}{64}a^{23}+\frac{27}{64}a^{22}-\frac{35}{128}a^{21}+\frac{1}{16}a^{20}-\frac{11}{64}a^{19}-\frac{5}{16}a^{18}+\frac{89}{512}a^{17}+\frac{173}{512}a^{16}-\frac{103}{512}a^{15}+\frac{29}{64}a^{14}+\frac{241}{512}a^{13}+\frac{11}{64}a^{12}+\frac{13}{256}a^{11}-\frac{5}{64}a^{10}-\frac{157}{512}a^{9}-\frac{121}{256}a^{8}-\frac{5}{128}a^{7}+\frac{27}{64}a^{6}-\frac{9}{32}a^{5}+\frac{3}{16}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{1024}a^{33}-\frac{1}{1024}a^{32}-\frac{1}{1024}a^{30}-\frac{1}{128}a^{29}+\frac{3}{512}a^{28}-\frac{3}{128}a^{27}+\frac{27}{1024}a^{26}-\frac{15}{512}a^{25}+\frac{7}{128}a^{24}+\frac{27}{128}a^{23}-\frac{35}{256}a^{22}+\frac{1}{32}a^{21}-\frac{11}{128}a^{20}-\frac{5}{32}a^{19}+\frac{89}{1024}a^{18}-\frac{339}{1024}a^{17}+\frac{409}{1024}a^{16}-\frac{35}{128}a^{15}-\frac{271}{1024}a^{14}+\frac{11}{128}a^{13}+\frac{13}{512}a^{12}+\frac{59}{128}a^{11}+\frac{355}{1024}a^{10}+\frac{135}{512}a^{9}-\frac{5}{256}a^{8}+\frac{27}{128}a^{7}+\frac{23}{64}a^{6}-\frac{13}{32}a^{5}+\frac{5}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2048}a^{34}-\frac{1}{2048}a^{33}-\frac{1}{2048}a^{31}-\frac{1}{256}a^{30}+\frac{3}{1024}a^{29}-\frac{3}{256}a^{28}+\frac{27}{2048}a^{27}-\frac{15}{1024}a^{26}+\frac{7}{256}a^{25}+\frac{27}{256}a^{24}-\frac{35}{512}a^{23}-\frac{31}{64}a^{22}+\frac{117}{256}a^{21}-\frac{5}{64}a^{20}-\frac{935}{2048}a^{19}-\frac{339}{2048}a^{18}+\frac{409}{2048}a^{17}-\frac{35}{256}a^{16}-\frac{271}{2048}a^{15}+\frac{11}{256}a^{14}-\frac{499}{1024}a^{13}+\frac{59}{256}a^{12}-\frac{669}{2048}a^{11}-\frac{377}{1024}a^{10}-\frac{5}{512}a^{9}+\frac{27}{256}a^{8}-\frac{41}{128}a^{7}-\frac{13}{64}a^{6}-\frac{11}{32}a^{5}-\frac{3}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4096}a^{35}-\frac{1}{4096}a^{34}-\frac{1}{4096}a^{32}-\frac{1}{512}a^{31}+\frac{3}{2048}a^{30}-\frac{3}{512}a^{29}+\frac{27}{4096}a^{28}-\frac{15}{2048}a^{27}+\frac{7}{512}a^{26}+\frac{27}{512}a^{25}-\frac{35}{1024}a^{24}+\frac{33}{128}a^{23}-\frac{139}{512}a^{22}+\frac{59}{128}a^{21}+\frac{1113}{4096}a^{20}+\frac{1709}{4096}a^{19}-\frac{1639}{4096}a^{18}-\frac{35}{512}a^{17}+\frac{1777}{4096}a^{16}+\frac{11}{512}a^{15}+\frac{525}{2048}a^{14}+\frac{59}{512}a^{13}+\frac{1379}{4096}a^{12}-\frac{377}{2048}a^{11}+\frac{507}{1024}a^{10}-\frac{229}{512}a^{9}+\frac{87}{256}a^{8}-\frac{13}{128}a^{7}-\frac{11}{64}a^{6}-\frac{3}{32}a^{5}-\frac{3}{8}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8192}a^{36}-\frac{1}{8192}a^{35}-\frac{1}{8192}a^{33}-\frac{1}{1024}a^{32}+\frac{3}{4096}a^{31}-\frac{3}{1024}a^{30}+\frac{27}{8192}a^{29}-\frac{15}{4096}a^{28}+\frac{7}{1024}a^{27}+\frac{27}{1024}a^{26}-\frac{35}{2048}a^{25}+\frac{33}{256}a^{24}-\frac{139}{1024}a^{23}+\frac{59}{256}a^{22}-\frac{2983}{8192}a^{21}-\frac{2387}{8192}a^{20}-\frac{1639}{8192}a^{19}+\frac{477}{1024}a^{18}-\frac{2319}{8192}a^{17}-\frac{501}{1024}a^{16}-\frac{1523}{4096}a^{15}-\frac{453}{1024}a^{14}-\frac{2717}{8192}a^{13}+\frac{1671}{4096}a^{12}+\frac{507}{2048}a^{11}-\frac{229}{1024}a^{10}+\frac{87}{512}a^{9}-\frac{13}{256}a^{8}+\frac{53}{128}a^{7}+\frac{29}{64}a^{6}-\frac{3}{16}a^{5}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16384}a^{37}-\frac{1}{16384}a^{36}-\frac{1}{16384}a^{34}-\frac{1}{2048}a^{33}+\frac{3}{8192}a^{32}-\frac{3}{2048}a^{31}+\frac{27}{16384}a^{30}-\frac{15}{8192}a^{29}+\frac{7}{2048}a^{28}+\frac{27}{2048}a^{27}-\frac{35}{4096}a^{26}+\frac{33}{512}a^{25}-\frac{139}{2048}a^{24}-\frac{197}{512}a^{23}+\frac{5209}{16384}a^{22}+\frac{5805}{16384}a^{21}-\frac{1639}{16384}a^{20}+\frac{477}{2048}a^{19}-\frac{2319}{16384}a^{18}-\frac{501}{2048}a^{17}+\frac{2573}{8192}a^{16}-\frac{453}{2048}a^{15}-\frac{2717}{16384}a^{14}-\frac{2425}{8192}a^{13}+\frac{507}{4096}a^{12}+\frac{795}{2048}a^{11}-\frac{425}{1024}a^{10}+\frac{243}{512}a^{9}-\frac{75}{256}a^{8}-\frac{35}{128}a^{7}-\frac{3}{32}a^{6}-\frac{7}{16}a^{5}+\frac{1}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{32768}a^{38}-\frac{1}{32768}a^{37}-\frac{1}{32768}a^{35}-\frac{1}{4096}a^{34}+\frac{3}{16384}a^{33}-\frac{3}{4096}a^{32}+\frac{27}{32768}a^{31}-\frac{15}{16384}a^{30}+\frac{7}{4096}a^{29}+\frac{27}{4096}a^{28}-\frac{35}{8192}a^{27}+\frac{33}{1024}a^{26}-\frac{139}{4096}a^{25}-\frac{197}{1024}a^{24}+\frac{5209}{32768}a^{23}-\frac{10579}{32768}a^{22}+\frac{14745}{32768}a^{21}-\frac{1571}{4096}a^{20}-\frac{2319}{32768}a^{19}-\frac{501}{4096}a^{18}-\frac{5619}{16384}a^{17}-\frac{453}{4096}a^{16}-\frac{2717}{32768}a^{15}-\frac{2425}{16384}a^{14}-\frac{3589}{8192}a^{13}-\frac{1253}{4096}a^{12}-\frac{425}{2048}a^{11}-\frac{269}{1024}a^{10}-\frac{75}{512}a^{9}-\frac{35}{256}a^{8}-\frac{3}{64}a^{7}+\frac{9}{32}a^{6}+\frac{1}{32}a^{5}+\frac{5}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{65536}a^{39}-\frac{1}{65536}a^{38}-\frac{1}{65536}a^{36}-\frac{1}{8192}a^{35}+\frac{3}{32768}a^{34}-\frac{3}{8192}a^{33}+\frac{27}{65536}a^{32}-\frac{15}{32768}a^{31}+\frac{7}{8192}a^{30}+\frac{27}{8192}a^{29}-\frac{35}{16384}a^{28}+\frac{33}{2048}a^{27}-\frac{139}{8192}a^{26}-\frac{197}{2048}a^{25}+\frac{5209}{65536}a^{24}-\frac{10579}{65536}a^{23}+\frac{14745}{65536}a^{22}+\frac{2525}{8192}a^{21}-\frac{2319}{65536}a^{20}-\frac{501}{8192}a^{19}+\frac{10765}{32768}a^{18}+\frac{3643}{8192}a^{17}+\frac{30051}{65536}a^{16}+\frac{13959}{32768}a^{15}-\frac{3589}{16384}a^{14}+\frac{2843}{8192}a^{13}+\frac{1623}{4096}a^{12}+\frac{755}{2048}a^{11}+\frac{437}{1024}a^{10}-\frac{35}{512}a^{9}+\frac{61}{128}a^{8}+\frac{9}{64}a^{7}+\frac{1}{64}a^{6}-\frac{11}{32}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{131072}a^{40}-\frac{1}{131072}a^{39}-\frac{1}{131072}a^{37}-\frac{1}{16384}a^{36}+\frac{3}{65536}a^{35}-\frac{3}{16384}a^{34}+\frac{27}{131072}a^{33}-\frac{15}{65536}a^{32}+\frac{7}{16384}a^{31}+\frac{27}{16384}a^{30}-\frac{35}{32768}a^{29}+\frac{33}{4096}a^{28}-\frac{139}{16384}a^{27}-\frac{197}{4096}a^{26}+\frac{5209}{131072}a^{25}-\frac{10579}{131072}a^{24}+\frac{14745}{131072}a^{23}+\frac{2525}{16384}a^{22}-\frac{2319}{131072}a^{21}-\frac{501}{16384}a^{20}+\frac{10765}{65536}a^{19}-\frac{4549}{16384}a^{18}-\frac{35485}{131072}a^{17}-\frac{18809}{65536}a^{16}-\frac{3589}{32768}a^{15}+\frac{2843}{16384}a^{14}-\frac{2473}{8192}a^{13}-\frac{1293}{4096}a^{12}-\frac{587}{2048}a^{11}-\frac{35}{1024}a^{10}-\frac{67}{256}a^{9}+\frac{9}{128}a^{8}-\frac{63}{128}a^{7}+\frac{21}{64}a^{6}+\frac{7}{16}a^{5}-\frac{3}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{262144}a^{41}-\frac{1}{262144}a^{40}-\frac{1}{262144}a^{38}-\frac{1}{32768}a^{37}+\frac{3}{131072}a^{36}-\frac{3}{32768}a^{35}+\frac{27}{262144}a^{34}-\frac{15}{131072}a^{33}+\frac{7}{32768}a^{32}+\frac{27}{32768}a^{31}-\frac{35}{65536}a^{30}+\frac{33}{8192}a^{29}-\frac{139}{32768}a^{28}-\frac{197}{8192}a^{27}+\frac{5209}{262144}a^{26}-\frac{10579}{262144}a^{25}+\frac{14745}{262144}a^{24}-\frac{13859}{32768}a^{23}+\frac{128753}{262144}a^{22}+\frac{15883}{32768}a^{21}+\frac{10765}{131072}a^{20}-\frac{4549}{32768}a^{19}-\frac{35485}{262144}a^{18}+\frac{46727}{131072}a^{17}-\frac{3589}{65536}a^{16}-\frac{13541}{32768}a^{15}-\frac{2473}{16384}a^{14}-\frac{1293}{8192}a^{13}-\frac{587}{4096}a^{12}-\frac{35}{2048}a^{11}+\frac{189}{512}a^{10}+\frac{9}{256}a^{9}-\frac{63}{256}a^{8}+\frac{21}{128}a^{7}-\frac{9}{32}a^{6}+\frac{5}{16}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{524288}a^{42}-\frac{1}{524288}a^{41}-\frac{1}{524288}a^{39}-\frac{1}{65536}a^{38}+\frac{3}{262144}a^{37}-\frac{3}{65536}a^{36}+\frac{27}{524288}a^{35}-\frac{15}{262144}a^{34}+\frac{7}{65536}a^{33}+\frac{27}{65536}a^{32}-\frac{35}{131072}a^{31}+\frac{33}{16384}a^{30}-\frac{139}{65536}a^{29}-\frac{197}{16384}a^{28}+\frac{5209}{524288}a^{27}-\frac{10579}{524288}a^{26}+\frac{14745}{524288}a^{25}-\frac{13859}{65536}a^{24}+\frac{128753}{524288}a^{23}+\frac{15883}{65536}a^{22}+\frac{10765}{262144}a^{21}+\frac{28219}{65536}a^{20}-\frac{35485}{524288}a^{19}-\frac{84345}{262144}a^{18}-\frac{3589}{131072}a^{17}+\frac{19227}{65536}a^{16}+\frac{13911}{32768}a^{15}-\frac{1293}{16384}a^{14}+\frac{3509}{8192}a^{13}-\frac{35}{4096}a^{12}-\frac{323}{1024}a^{11}-\frac{247}{512}a^{10}-\frac{63}{512}a^{9}+\frac{21}{256}a^{8}+\frac{23}{64}a^{7}-\frac{11}{32}a^{6}+\frac{7}{16}a^{5}-\frac{3}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{1048576}a^{43}-\frac{1}{1048576}a^{42}-\frac{1}{1048576}a^{40}-\frac{1}{131072}a^{39}+\frac{3}{524288}a^{38}-\frac{3}{131072}a^{37}+\frac{27}{1048576}a^{36}-\frac{15}{524288}a^{35}+\frac{7}{131072}a^{34}+\frac{27}{131072}a^{33}-\frac{35}{262144}a^{32}+\frac{33}{32768}a^{31}-\frac{139}{131072}a^{30}-\frac{197}{32768}a^{29}+\frac{5209}{1048576}a^{28}-\frac{10579}{1048576}a^{27}+\frac{14745}{1048576}a^{26}-\frac{13859}{131072}a^{25}+\frac{128753}{1048576}a^{24}-\frac{49653}{131072}a^{23}-\frac{251379}{524288}a^{22}-\frac{37317}{131072}a^{21}-\frac{35485}{1048576}a^{20}-\frac{84345}{524288}a^{19}-\frac{3589}{262144}a^{18}+\frac{19227}{131072}a^{17}-\frac{18857}{65536}a^{16}-\frac{1293}{32768}a^{15}-\frac{4683}{16384}a^{14}+\frac{4061}{8192}a^{13}+\frac{701}{2048}a^{12}-\frac{247}{1024}a^{11}-\frac{63}{1024}a^{10}-\frac{235}{512}a^{9}-\frac{41}{128}a^{8}+\frac{21}{64}a^{7}-\frac{9}{32}a^{6}+\frac{5}{16}a^{5}-\frac{5}{16}a^{4}+\frac{3}{8}a^{3}$, $\frac{1}{589299712}a^{44}+\frac{171}{589299712}a^{43}+\frac{55}{73662464}a^{42}-\frac{821}{589299712}a^{41}-\frac{445}{147324928}a^{40}+\frac{889}{294649856}a^{39}-\frac{99}{9207808}a^{38}+\frac{6307}{589299712}a^{37}+\frac{15571}{294649856}a^{36}+\frac{6943}{147324928}a^{35}-\frac{6783}{36831232}a^{34}+\frac{47465}{147324928}a^{33}-\frac{35637}{36831232}a^{32}+\frac{89807}{73662464}a^{31}+\frac{7201}{4603904}a^{30}+\frac{4120377}{589299712}a^{29}+\frac{8550905}{589299712}a^{28}+\frac{17476377}{589299712}a^{27}+\frac{996415}{73662464}a^{26}+\frac{42481013}{589299712}a^{25}-\frac{21227943}{147324928}a^{24}-\frac{18088361}{294649856}a^{23}+\frac{3582625}{9207808}a^{22}+\frac{199770203}{589299712}a^{21}-\frac{24535303}{294649856}a^{20}-\frac{5945583}{18415616}a^{19}-\frac{7803053}{36831232}a^{18}-\frac{6853973}{36831232}a^{17}+\frac{1921073}{9207808}a^{16}-\frac{3816321}{9207808}a^{15}-\frac{215165}{4603904}a^{14}-\frac{134409}{2301952}a^{13}+\frac{197445}{1150976}a^{12}-\frac{3547}{17984}a^{11}+\frac{82337}{287744}a^{10}-\frac{25863}{71936}a^{9}+\frac{3141}{35968}a^{8}+\frac{6265}{35968}a^{7}+\frac{6899}{17984}a^{6}-\frac{1683}{8992}a^{5}-\frac{831}{4496}a^{4}-\frac{159}{2248}a^{3}+\frac{19}{562}a^{2}+\frac{21}{281}a-\frac{55}{281}$, $\frac{1}{20\!\cdots\!64}a^{45}+\frac{15\!\cdots\!41}{20\!\cdots\!64}a^{44}+\frac{15\!\cdots\!13}{10\!\cdots\!32}a^{43}+\frac{82\!\cdots\!55}{20\!\cdots\!64}a^{42}-\frac{83\!\cdots\!49}{10\!\cdots\!32}a^{41}+\frac{40\!\cdots\!41}{10\!\cdots\!32}a^{40}+\frac{21\!\cdots\!31}{50\!\cdots\!16}a^{39}-\frac{12\!\cdots\!65}{20\!\cdots\!64}a^{38}-\frac{17\!\cdots\!05}{25\!\cdots\!08}a^{37}-\frac{27\!\cdots\!75}{12\!\cdots\!04}a^{36}-\frac{17\!\cdots\!51}{25\!\cdots\!08}a^{35}-\frac{93\!\cdots\!43}{50\!\cdots\!16}a^{34}+\frac{46\!\cdots\!73}{25\!\cdots\!08}a^{33}-\frac{20\!\cdots\!49}{25\!\cdots\!08}a^{32}-\frac{89\!\cdots\!99}{12\!\cdots\!04}a^{31}+\frac{24\!\cdots\!49}{20\!\cdots\!64}a^{30}+\frac{31\!\cdots\!03}{20\!\cdots\!64}a^{29}+\frac{20\!\cdots\!99}{20\!\cdots\!64}a^{28}-\frac{31\!\cdots\!29}{10\!\cdots\!32}a^{27}-\frac{67\!\cdots\!19}{20\!\cdots\!64}a^{26}+\frac{32\!\cdots\!81}{10\!\cdots\!32}a^{25}-\frac{13\!\cdots\!29}{10\!\cdots\!32}a^{24}+\frac{21\!\cdots\!49}{50\!\cdots\!16}a^{23}-\frac{38\!\cdots\!49}{20\!\cdots\!64}a^{22}-\frac{12\!\cdots\!75}{50\!\cdots\!16}a^{21}-\frac{26\!\cdots\!37}{50\!\cdots\!16}a^{20}-\frac{11\!\cdots\!17}{31\!\cdots\!76}a^{19}+\frac{25\!\cdots\!15}{15\!\cdots\!88}a^{18}-\frac{11\!\cdots\!33}{63\!\cdots\!52}a^{17}+\frac{41\!\cdots\!91}{15\!\cdots\!88}a^{16}-\frac{63\!\cdots\!37}{24\!\cdots\!92}a^{15}+\frac{15\!\cdots\!47}{39\!\cdots\!72}a^{14}-\frac{72\!\cdots\!35}{39\!\cdots\!72}a^{13}+\frac{27\!\cdots\!55}{19\!\cdots\!36}a^{12}+\frac{21\!\cdots\!45}{49\!\cdots\!84}a^{11}-\frac{11\!\cdots\!31}{49\!\cdots\!84}a^{10}-\frac{10\!\cdots\!37}{24\!\cdots\!92}a^{9}+\frac{81\!\cdots\!17}{15\!\cdots\!12}a^{8}+\frac{75\!\cdots\!77}{62\!\cdots\!48}a^{7}+\frac{33\!\cdots\!29}{15\!\cdots\!12}a^{6}-\frac{15\!\cdots\!27}{77\!\cdots\!56}a^{5}+\frac{29\!\cdots\!59}{77\!\cdots\!56}a^{4}+\frac{68\!\cdots\!17}{38\!\cdots\!28}a^{3}-\frac{29\!\cdots\!97}{97\!\cdots\!82}a^{2}-\frac{93\!\cdots\!03}{97\!\cdots\!82}a+\frac{23\!\cdots\!28}{48\!\cdots\!41}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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:  $22$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 + 68*x^44 - 61*x^43 + 2272*x^42 - 1838*x^41 + 49088*x^40 - 35837*x^39 + 764082*x^38 - 502716*x^37 + 9061304*x^36 - 5356876*x^35 + 84584992*x^34 - 44739144*x^33 + 634011776*x^32 - 298356295*x^31 + 3862036333*x^30 - 1605788227*x^29 + 19242860452*x^28 - 7010057107*x^27 + 78601898736*x^26 - 24835843890*x^25 + 262854128576*x^24 - 71169036677*x^23 + 716241923902*x^22 - 163773410984*x^21 + 1577302832552*x^20 - 299245739680*x^19 + 2773773179200*x^18 - 427289762048*x^17 + 3831228318848*x^16 - 466369885184*x^15 + 4064622286848*x^14 - 377834643456*x^13 + 3213707960320*x^12 - 217643319296*x^11 + 1816862654464*x^10 - 84719697920*x^9 + 692213350400*x^8 - 19502399488*x^7 + 162748039168*x^6 - 3073900544*x^5 + 20306198528*x^4 + 161480704*x^3 + 1019215872*x^2 - 50331648*x + 8388608)
 
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 - x^45 + 68*x^44 - 61*x^43 + 2272*x^42 - 1838*x^41 + 49088*x^40 - 35837*x^39 + 764082*x^38 - 502716*x^37 + 9061304*x^36 - 5356876*x^35 + 84584992*x^34 - 44739144*x^33 + 634011776*x^32 - 298356295*x^31 + 3862036333*x^30 - 1605788227*x^29 + 19242860452*x^28 - 7010057107*x^27 + 78601898736*x^26 - 24835843890*x^25 + 262854128576*x^24 - 71169036677*x^23 + 716241923902*x^22 - 163773410984*x^21 + 1577302832552*x^20 - 299245739680*x^19 + 2773773179200*x^18 - 427289762048*x^17 + 3831228318848*x^16 - 466369885184*x^15 + 4064622286848*x^14 - 377834643456*x^13 + 3213707960320*x^12 - 217643319296*x^11 + 1816862654464*x^10 - 84719697920*x^9 + 692213350400*x^8 - 19502399488*x^7 + 162748039168*x^6 - 3073900544*x^5 + 20306198528*x^4 + 161480704*x^3 + 1019215872*x^2 - 50331648*x + 8388608, 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 - x^45 + 68*x^44 - 61*x^43 + 2272*x^42 - 1838*x^41 + 49088*x^40 - 35837*x^39 + 764082*x^38 - 502716*x^37 + 9061304*x^36 - 5356876*x^35 + 84584992*x^34 - 44739144*x^33 + 634011776*x^32 - 298356295*x^31 + 3862036333*x^30 - 1605788227*x^29 + 19242860452*x^28 - 7010057107*x^27 + 78601898736*x^26 - 24835843890*x^25 + 262854128576*x^24 - 71169036677*x^23 + 716241923902*x^22 - 163773410984*x^21 + 1577302832552*x^20 - 299245739680*x^19 + 2773773179200*x^18 - 427289762048*x^17 + 3831228318848*x^16 - 466369885184*x^15 + 4064622286848*x^14 - 377834643456*x^13 + 3213707960320*x^12 - 217643319296*x^11 + 1816862654464*x^10 - 84719697920*x^9 + 692213350400*x^8 - 19502399488*x^7 + 162748039168*x^6 - 3073900544*x^5 + 20306198528*x^4 + 161480704*x^3 + 1019215872*x^2 - 50331648*x + 8388608);
 
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 - x^45 + 68*x^44 - 61*x^43 + 2272*x^42 - 1838*x^41 + 49088*x^40 - 35837*x^39 + 764082*x^38 - 502716*x^37 + 9061304*x^36 - 5356876*x^35 + 84584992*x^34 - 44739144*x^33 + 634011776*x^32 - 298356295*x^31 + 3862036333*x^30 - 1605788227*x^29 + 19242860452*x^28 - 7010057107*x^27 + 78601898736*x^26 - 24835843890*x^25 + 262854128576*x^24 - 71169036677*x^23 + 716241923902*x^22 - 163773410984*x^21 + 1577302832552*x^20 - 299245739680*x^19 + 2773773179200*x^18 - 427289762048*x^17 + 3831228318848*x^16 - 466369885184*x^15 + 4064622286848*x^14 - 377834643456*x^13 + 3213707960320*x^12 - 217643319296*x^11 + 1816862654464*x^10 - 84719697920*x^9 + 692213350400*x^8 - 19502399488*x^7 + 162748039168*x^6 - 3073900544*x^5 + 20306198528*x^4 + 161480704*x^3 + 1019215872*x^2 - 50331648*x + 8388608);
 
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

$C_{46}$ (as 46T1):

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 cyclic group of order 46
The 46 conjugacy class representatives for $C_{46}$
Character table for $C_{46}$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{47})^+\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $23^{2}$ $46$ $46$ R $23^{2}$ $46$ $46$ $46$ $23^{2}$ $23^{2}$ $46$ $23^{2}$ $46$ $23^{2}$ R $23^{2}$ $46$

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]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display Deg $46$$2$$23$$23$
\(47\) Copy content Toggle raw display Deg $46$$23$$2$$44$