LMFDB - Number field 32.0.14680252274932068764306928624388060814210236416.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536)

gp: K = bnfinit(y^32 - 4*y^30 + 10*y^28 - 32*y^26 + 83*y^24 - 208*y^22 + 458*y^20 - 956*y^18 + 2113*y^16 - 3824*y^14 + 7328*y^12 - 13312*y^10 + 21248*y^8 - 32768*y^6 + 40960*y^4 - 65536*y^2 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536)

$ x^{32} - 4 x^{30} + 10 x^{28} - 32 x^{26} + 83 x^{24} - 208 x^{22} + 458 x^{20} - 956 x^{18} + \cdots + 65536 $ x^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$32$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 16]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$14680252274932068764306928624388060814210236416$ 14680252274932068764306928624388060814210236416 $\medspace = 2^{72}\cdot 3^{16}\cdot 13^{8}\cdot 97^{4}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$27.71$	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:	$2^{9/4}3^{1/2}13^{1/2}97^{1/2}\approx 292.57395390451063$
Ramified primes:	$2$, $3$, $13$, $97$ 2, 3, 13, 97	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{12}a^{16}+\frac{1}{12}a^{14}-\frac{1}{12}a^{10}+\frac{1}{6}a^{8}+\frac{5}{12}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{12}a^{17}+\frac{1}{12}a^{15}-\frac{1}{12}a^{11}+\frac{1}{6}a^{9}-\frac{1}{12}a^{7}-\frac{1}{4}a^{5}-\frac{1}{6}a^{3}-\frac{1}{6}a$, $\frac{1}{12}a^{18}-\frac{1}{12}a^{14}-\frac{1}{12}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{6}a^{6}-\frac{1}{2}a^{5}+\frac{1}{12}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{24}a^{19}+\frac{1}{12}a^{15}-\frac{1}{6}a^{13}+\frac{1}{8}a^{11}-\frac{1}{12}a^{7}-\frac{1}{2}a^{6}-\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a$, $\frac{1}{48}a^{20}-\frac{1}{24}a^{16}+\frac{1}{12}a^{14}+\frac{1}{16}a^{12}-\frac{1}{6}a^{10}+\frac{1}{24}a^{8}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}+\frac{7}{16}a^{4}-\frac{1}{2}a^{3}+\frac{1}{12}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{96}a^{21}+\frac{1}{48}a^{17}+\frac{1}{12}a^{15}-\frac{7}{32}a^{13}-\frac{1}{8}a^{11}-\frac{7}{48}a^{9}+\frac{1}{24}a^{7}-\frac{5}{32}a^{5}+\frac{11}{24}a^{3}-\frac{1}{2}a$, $\frac{1}{384}a^{22}-\frac{1}{192}a^{21}-\frac{1}{96}a^{20}+\frac{5}{192}a^{18}+\frac{1}{32}a^{17}-\frac{1}{8}a^{15}-\frac{13}{384}a^{14}+\frac{15}{64}a^{13}-\frac{5}{24}a^{12}-\frac{11}{48}a^{11}-\frac{43}{192}a^{10}+\frac{1}{32}a^{9}-\frac{5}{32}a^{8}+\frac{3}{16}a^{7}+\frac{97}{384}a^{6}-\frac{11}{64}a^{5}+\frac{7}{24}a^{4}-\frac{7}{16}a^{3}-\frac{1}{12}a^{2}+\frac{5}{12}a+\frac{1}{6}$, $\frac{1}{768}a^{23}-\frac{1}{96}a^{20}+\frac{5}{384}a^{19}+\frac{1}{96}a^{17}-\frac{1}{48}a^{16}-\frac{77}{768}a^{15}-\frac{1}{12}a^{14}+\frac{31}{192}a^{13}+\frac{7}{32}a^{12}-\frac{67}{384}a^{11}+\frac{1}{8}a^{10}-\frac{5}{192}a^{9}+\frac{7}{48}a^{8}-\frac{79}{768}a^{7}-\frac{1}{24}a^{6}+\frac{37}{192}a^{5}+\frac{5}{32}a^{4}-\frac{3}{16}a^{3}-\frac{11}{24}a^{2}+\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{1536}a^{24}-\frac{1}{192}a^{21}+\frac{5}{768}a^{20}-\frac{7}{192}a^{18}+\frac{1}{32}a^{17}-\frac{13}{1536}a^{16}-\frac{1}{8}a^{15}-\frac{11}{128}a^{14}+\frac{15}{64}a^{13}+\frac{157}{768}a^{12}-\frac{11}{48}a^{11}-\frac{23}{128}a^{10}+\frac{1}{32}a^{9}-\frac{143}{1536}a^{8}+\frac{3}{16}a^{7}-\frac{139}{384}a^{6}-\frac{11}{64}a^{5}+\frac{47}{96}a^{4}-\frac{7}{16}a^{3}-\frac{1}{6}a^{2}+\frac{5}{12}a+\frac{1}{3}$, $\frac{1}{3072}a^{25}-\frac{1}{512}a^{21}-\frac{1}{96}a^{20}-\frac{7}{384}a^{19}-\frac{15}{1024}a^{17}-\frac{1}{48}a^{16}-\frac{65}{768}a^{15}-\frac{1}{12}a^{14}+\frac{325}{1536}a^{13}+\frac{7}{32}a^{12}-\frac{7}{256}a^{11}+\frac{1}{8}a^{10}-\frac{229}{1024}a^{9}+\frac{7}{48}a^{8}+\frac{37}{768}a^{7}-\frac{1}{24}a^{6}+\frac{7}{96}a^{5}+\frac{5}{32}a^{4}+\frac{3}{16}a^{3}-\frac{11}{24}a^{2}-\frac{1}{12}a-\frac{1}{2}$, $\frac{1}{6144}a^{26}-\frac{1}{1024}a^{22}-\frac{1}{192}a^{21}-\frac{7}{768}a^{20}+\frac{211}{6144}a^{18}+\frac{1}{32}a^{17}-\frac{1}{1536}a^{16}-\frac{1}{8}a^{15}+\frac{325}{3072}a^{14}+\frac{15}{64}a^{13}+\frac{299}{1536}a^{12}-\frac{11}{48}a^{11}-\frac{175}{6144}a^{10}+\frac{1}{32}a^{9}+\frac{119}{512}a^{8}+\frac{3}{16}a^{7}-\frac{17}{192}a^{6}-\frac{11}{64}a^{5}-\frac{23}{96}a^{4}-\frac{7}{16}a^{3}-\frac{1}{8}a^{2}+\frac{5}{12}a$, $\frac{1}{12288}a^{27}-\frac{1}{2048}a^{23}+\frac{1}{1536}a^{21}-\frac{1}{96}a^{20}+\frac{211}{12288}a^{19}+\frac{31}{3072}a^{17}-\frac{1}{48}a^{16}+\frac{581}{6144}a^{15}-\frac{1}{12}a^{14}+\frac{731}{3072}a^{13}+\frac{7}{32}a^{12}+\frac{2129}{12288}a^{11}+\frac{1}{8}a^{10}+\frac{133}{3072}a^{9}+\frac{7}{48}a^{8}-\frac{3}{128}a^{7}+\frac{11}{24}a^{6}+\frac{29}{96}a^{5}-\frac{11}{32}a^{4}+\frac{5}{12}a^{3}+\frac{1}{24}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{19218432}a^{28}-\frac{1}{1201152}a^{26}+\frac{815}{3203072}a^{24}+\frac{2053}{2402304}a^{22}-\frac{1}{192}a^{21}-\frac{94573}{19218432}a^{20}-\frac{111245}{4804608}a^{18}+\frac{1}{32}a^{17}-\frac{199147}{9609216}a^{16}-\frac{1}{8}a^{15}-\frac{310685}{4804608}a^{14}+\frac{15}{64}a^{13}-\frac{1111855}{19218432}a^{12}-\frac{11}{48}a^{11}-\frac{788735}{4804608}a^{10}+\frac{1}{32}a^{9}-\frac{58601}{300288}a^{8}+\frac{3}{16}a^{7}-\frac{110645}{300288}a^{6}+\frac{21}{64}a^{5}+\frac{6845}{18768}a^{4}+\frac{1}{16}a^{3}+\frac{5801}{18768}a^{2}-\frac{1}{12}a-\frac{129}{1564}$, $\frac{1}{38436864}a^{29}-\frac{1}{2402304}a^{27}+\frac{815}{6406144}a^{25}+\frac{2053}{4804608}a^{23}+\frac{105619}{38436864}a^{21}-\frac{1}{96}a^{20}-\frac{111245}{9609216}a^{19}+\frac{1045}{19218432}a^{17}-\frac{1}{48}a^{16}+\frac{89699}{9609216}a^{15}-\frac{1}{12}a^{14}+\frac{4293329}{38436864}a^{13}+\frac{7}{32}a^{12}-\frac{1389311}{9609216}a^{11}+\frac{1}{8}a^{10}+\frac{15917}{200192}a^{9}+\frac{7}{48}a^{8}-\frac{32711}{200192}a^{7}-\frac{1}{24}a^{6}-\frac{10943}{75072}a^{5}+\frac{5}{32}a^{4}-\frac{1455}{12512}a^{3}-\frac{11}{24}a^{2}+\frac{653}{3128}a-\frac{1}{2}$, $\frac{1}{10992943104}a^{30}+\frac{35}{1374117888}a^{28}-\frac{12505}{499679232}a^{26}-\frac{111049}{1374117888}a^{24}-\frac{11198381}{10992943104}a^{22}-\frac{1}{192}a^{21}+\frac{17488945}{2748235776}a^{20}+\frac{74336183}{1832157184}a^{18}+\frac{1}{32}a^{17}+\frac{4961383}{2748235776}a^{16}-\frac{1}{8}a^{15}-\frac{15625}{477954048}a^{14}+\frac{15}{64}a^{13}-\frac{167246741}{2748235776}a^{12}-\frac{11}{48}a^{11}+\frac{29578729}{343529472}a^{10}+\frac{1}{32}a^{9}+\frac{10835437}{171764736}a^{8}+\frac{3}{16}a^{7}+\frac{172087}{447304}a^{6}-\frac{11}{64}a^{5}-\frac{447989}{975936}a^{4}-\frac{7}{16}a^{3}-\frac{868451}{2683824}a^{2}+\frac{5}{12}a+\frac{50122}{167739}$, $\frac{1}{21985886208}a^{31}+\frac{35}{2748235776}a^{29}-\frac{12505}{999358464}a^{27}-\frac{111049}{2748235776}a^{25}-\frac{11198381}{21985886208}a^{23}-\frac{3712837}{1832157184}a^{21}-\frac{1}{96}a^{20}+\frac{74336183}{3664314368}a^{19}+\frac{176726119}{5496471552}a^{17}-\frac{1}{48}a^{16}-\frac{15625}{955908096}a^{15}-\frac{1}{12}a^{14}+\frac{433929835}{5496471552}a^{13}+\frac{7}{32}a^{12}-\frac{42624093}{229019648}a^{11}+\frac{1}{8}a^{10}+\frac{64511917}{343529472}a^{9}+\frac{7}{48}a^{8}+\frac{58087}{447304}a^{7}+\frac{11}{24}a^{6}+\frac{436453}{1951872}a^{5}-\frac{11}{32}a^{4}-\frac{1203929}{5367648}a^{3}+\frac{1}{24}a^{2}+\frac{14777}{223652}a-\frac{1}{2}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/2*a^7 - 1/2*a^6 - 1/2*a^4 - 1/2*a^2 - 1/2*a, 1/2*a^8 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a, 1/2*a^9 - 1/2*a^5 - 1/2*a, 1/2*a^10 - 1/2*a^6 - 1/2*a^2, 1/2*a^11 - 1/2*a^6 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/2*a^12 - 1/2*a^6 - 1/2*a^5 - 1/2*a^3 - 1/2*a, 1/2*a^13 - 1/2*a, 1/4*a^14 - 1/4*a^12 - 1/4*a^8 - 1/4*a^4 - 1/4*a^2, 1/4*a^15 - 1/4*a^13 - 1/4*a^9 - 1/4*a^5 - 1/4*a^3, 1/12*a^16 + 1/12*a^14 - 1/12*a^10 + 1/6*a^8 + 5/12*a^6 - 1/2*a^5 - 1/4*a^4 - 1/2*a^3 - 1/6*a^2 - 1/2*a + 1/3, 1/12*a^17 + 1/12*a^15 - 1/12*a^11 + 1/6*a^9 - 1/12*a^7 - 1/4*a^5 - 1/6*a^3 - 1/6*a, 1/12*a^18 - 1/12*a^14 - 1/12*a^12 - 1/4*a^10 - 1/4*a^8 - 1/6*a^6 - 1/2*a^5 + 1/12*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 1/3, 1/24*a^19 + 1/12*a^15 - 1/6*a^13 + 1/8*a^11 - 1/12*a^7 - 1/2*a^6 - 1/3*a^5 - 1/2*a^4 + 3/8*a^3 - 1/2*a^2 + 1/3*a, 1/48*a^20 - 1/24*a^16 + 1/12*a^14 + 1/16*a^12 - 1/6*a^10 + 1/24*a^8 - 1/3*a^6 - 1/2*a^5 + 7/16*a^4 - 1/2*a^3 + 1/12*a^2 - 1/2*a - 1/3, 1/96*a^21 + 1/48*a^17 + 1/12*a^15 - 7/32*a^13 - 1/8*a^11 - 7/48*a^9 + 1/24*a^7 - 5/32*a^5 + 11/24*a^3 - 1/2*a, 1/384*a^22 - 1/192*a^21 - 1/96*a^20 + 5/192*a^18 + 1/32*a^17 - 1/8*a^15 - 13/384*a^14 + 15/64*a^13 - 5/24*a^12 - 11/48*a^11 - 43/192*a^10 + 1/32*a^9 - 5/32*a^8 + 3/16*a^7 + 97/384*a^6 - 11/64*a^5 + 7/24*a^4 - 7/16*a^3 - 1/12*a^2 + 5/12*a + 1/6, 1/768*a^23 - 1/96*a^20 + 5/384*a^19 + 1/96*a^17 - 1/48*a^16 - 77/768*a^15 - 1/12*a^14 + 31/192*a^13 + 7/32*a^12 - 67/384*a^11 + 1/8*a^10 - 5/192*a^9 + 7/48*a^8 - 79/768*a^7 - 1/24*a^6 + 37/192*a^5 + 5/32*a^4 - 3/16*a^3 - 11/24*a^2 + 1/3*a - 1/2, 1/1536*a^24 - 1/192*a^21 + 5/768*a^20 - 7/192*a^18 + 1/32*a^17 - 13/1536*a^16 - 1/8*a^15 - 11/128*a^14 + 15/64*a^13 + 157/768*a^12 - 11/48*a^11 - 23/128*a^10 + 1/32*a^9 - 143/1536*a^8 + 3/16*a^7 - 139/384*a^6 - 11/64*a^5 + 47/96*a^4 - 7/16*a^3 - 1/6*a^2 + 5/12*a + 1/3, 1/3072*a^25 - 1/512*a^21 - 1/96*a^20 - 7/384*a^19 - 15/1024*a^17 - 1/48*a^16 - 65/768*a^15 - 1/12*a^14 + 325/1536*a^13 + 7/32*a^12 - 7/256*a^11 + 1/8*a^10 - 229/1024*a^9 + 7/48*a^8 + 37/768*a^7 - 1/24*a^6 + 7/96*a^5 + 5/32*a^4 + 3/16*a^3 - 11/24*a^2 - 1/12*a - 1/2, 1/6144*a^26 - 1/1024*a^22 - 1/192*a^21 - 7/768*a^20 + 211/6144*a^18 + 1/32*a^17 - 1/1536*a^16 - 1/8*a^15 + 325/3072*a^14 + 15/64*a^13 + 299/1536*a^12 - 11/48*a^11 - 175/6144*a^10 + 1/32*a^9 + 119/512*a^8 + 3/16*a^7 - 17/192*a^6 - 11/64*a^5 - 23/96*a^4 - 7/16*a^3 - 1/8*a^2 + 5/12*a, 1/12288*a^27 - 1/2048*a^23 + 1/1536*a^21 - 1/96*a^20 + 211/12288*a^19 + 31/3072*a^17 - 1/48*a^16 + 581/6144*a^15 - 1/12*a^14 + 731/3072*a^13 + 7/32*a^12 + 2129/12288*a^11 + 1/8*a^10 + 133/3072*a^9 + 7/48*a^8 - 3/128*a^7 + 11/24*a^6 + 29/96*a^5 - 11/32*a^4 + 5/12*a^3 + 1/24*a^2 + 1/4*a - 1/2, 1/19218432*a^28 - 1/1201152*a^26 + 815/3203072*a^24 + 2053/2402304*a^22 - 1/192*a^21 - 94573/19218432*a^20 - 111245/4804608*a^18 + 1/32*a^17 - 199147/9609216*a^16 - 1/8*a^15 - 310685/4804608*a^14 + 15/64*a^13 - 1111855/19218432*a^12 - 11/48*a^11 - 788735/4804608*a^10 + 1/32*a^9 - 58601/300288*a^8 + 3/16*a^7 - 110645/300288*a^6 + 21/64*a^5 + 6845/18768*a^4 + 1/16*a^3 + 5801/18768*a^2 - 1/12*a - 129/1564, 1/38436864*a^29 - 1/2402304*a^27 + 815/6406144*a^25 + 2053/4804608*a^23 + 105619/38436864*a^21 - 1/96*a^20 - 111245/9609216*a^19 + 1045/19218432*a^17 - 1/48*a^16 + 89699/9609216*a^15 - 1/12*a^14 + 4293329/38436864*a^13 + 7/32*a^12 - 1389311/9609216*a^11 + 1/8*a^10 + 15917/200192*a^9 + 7/48*a^8 - 32711/200192*a^7 - 1/24*a^6 - 10943/75072*a^5 + 5/32*a^4 - 1455/12512*a^3 - 11/24*a^2 + 653/3128*a - 1/2, 1/10992943104*a^30 + 35/1374117888*a^28 - 12505/499679232*a^26 - 111049/1374117888*a^24 - 11198381/10992943104*a^22 - 1/192*a^21 + 17488945/2748235776*a^20 + 74336183/1832157184*a^18 + 1/32*a^17 + 4961383/2748235776*a^16 - 1/8*a^15 - 15625/477954048*a^14 + 15/64*a^13 - 167246741/2748235776*a^12 - 11/48*a^11 + 29578729/343529472*a^10 + 1/32*a^9 + 10835437/171764736*a^8 + 3/16*a^7 + 172087/447304*a^6 - 11/64*a^5 - 447989/975936*a^4 - 7/16*a^3 - 868451/2683824*a^2 + 5/12*a + 50122/167739, 1/21985886208*a^31 + 35/2748235776*a^29 - 12505/999358464*a^27 - 111049/2748235776*a^25 - 11198381/21985886208*a^23 - 3712837/1832157184*a^21 - 1/96*a^20 + 74336183/3664314368*a^19 + 176726119/5496471552*a^17 - 1/48*a^16 - 15625/955908096*a^15 - 1/12*a^14 + 433929835/5496471552*a^13 + 7/32*a^12 - 42624093/229019648*a^11 + 1/8*a^10 + 64511917/343529472*a^9 + 7/48*a^8 + 58087/447304*a^7 + 11/24*a^6 + 436453/1951872*a^5 - 11/32*a^4 - 1203929/5367648*a^3 + 1/24*a^2 + 14777/223652*a - 1/2

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{4}$, which has order $4$ (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:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{1215085}{10992943104} a^{31} - \frac{1394149}{5496471552} a^{29} + \frac{309323}{499679232} a^{27} - \frac{97963}{39829504} a^{25} + \frac{18384557}{3664314368} a^{23} - \frac{26777331}{1832157184} a^{21} + \frac{8784421}{323321856} a^{19} - \frac{82237741}{1374117888} a^{17} + \frac{1497652933}{10992943104} a^{15} - \frac{1103988403}{5496471552} a^{13} + \frac{220163729}{458039296} a^{11} - \frac{118625219}{171764736} a^{9} + \frac{104765123}{85882368} a^{7} - \frac{1638191}{975936} a^{5} + \frac{2897783}{1789216} a^{3} - \frac{5762987}{1341912} a $ (1215085)/(10992943104)a^(31) - (1394149)/(5496471552)a^(29) + (309323)/(499679232)a^(27) - (97963)/(39829504)a^(25) + (18384557)/(3664314368)a^(23) - (26777331)/(1832157184)a^(21) + (8784421)/(323321856)a^(19) - (82237741)/(1374117888)a^(17) + (1497652933)/(10992943104)a^(15) - (1103988403)/(5496471552)a^(13) + (220163729)/(458039296)a^(11) - (118625219)/(171764736)a^(9) + (104765123)/(85882368)a^(7) - (1638191)/(975936)a^(5) + (2897783)/(1789216)a^(3) - (5762987)/(1341912)a (order $24$)	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:	$\frac{598823}{5496471552}a^{30}-\frac{351331}{1374117888}a^{28}+\frac{152897}{249839616}a^{26}-\frac{112259}{42941184}a^{24}+\frac{28356293}{5496471552}a^{22}-\frac{656639}{42941184}a^{20}+\frac{25178825}{916078592}a^{18}-\frac{87450241}{1374117888}a^{16}+\frac{252322157}{1832157184}a^{14}-\frac{17461369}{85882368}a^{12}+\frac{58833033}{114509824}a^{10}-\frac{15680039}{21470592}a^{8}+\frac{9091837}{7156864}a^{6}-\frac{421673}{243984}a^{4}+\frac{127086}{55913}a^{2}-\frac{18737}{3289}$, $\frac{1215085}{10992943104}a^{31}-\frac{1394149}{5496471552}a^{29}+\frac{309323}{499679232}a^{27}-\frac{97963}{39829504}a^{25}+\frac{18384557}{3664314368}a^{23}-\frac{26777331}{1832157184}a^{21}+\frac{8784421}{323321856}a^{19}-\frac{82237741}{1374117888}a^{17}+\frac{1497652933}{10992943104}a^{15}-\frac{1103988403}{5496471552}a^{13}+\frac{220163729}{458039296}a^{11}-\frac{118625219}{171764736}a^{9}+\frac{104765123}{85882368}a^{7}-\frac{1638191}{975936}a^{5}+\frac{2897783}{1789216}a^{3}-\frac{5762987}{1341912}a-1$, $\frac{1468835}{7328628736}a^{31}-\frac{689955}{1832157184}a^{29}+\frac{1209311}{999358464}a^{27}-\frac{116087}{29872128}a^{25}+\frac{186816811}{21985886208}a^{23}-\frac{11159727}{458039296}a^{21}+\frac{9009127}{215547904}a^{19}-\frac{190415389}{1832157184}a^{17}+\frac{4584444361}{21985886208}a^{15}-\frac{473411611}{1374117888}a^{13}+\frac{1066638187}{1374117888}a^{11}-\frac{376994399}{343529472}a^{9}+\frac{58311373}{28627456}a^{7}-\frac{4761553}{1951872}a^{5}+\frac{2879671}{894608}a^{3}-\frac{8573359}{1341912}a-1$, $\frac{5280841}{21985886208}a^{31}-\frac{875341}{5496471552}a^{30}-\frac{589265}{1374117888}a^{29}+\frac{287897}{916078592}a^{28}+\frac{1334431}{999358464}a^{27}-\frac{219595}{249839616}a^{26}-\frac{12673747}{2748235776}a^{25}+\frac{4626629}{1374117888}a^{24}+\frac{71284601}{7328628736}a^{23}-\frac{2384983}{323321856}a^{22}-\frac{51928967}{1832157184}a^{21}+\frac{58607275}{2748235776}a^{20}+\frac{176210079}{3664314368}a^{19}-\frac{34565103}{916078592}a^{18}-\frac{221641559}{1832157184}a^{17}+\frac{16259467}{171764736}a^{16}+\frac{5212804121}{21985886208}a^{15}-\frac{1107454741}{5496471552}a^{14}-\frac{90750593}{238977024}a^{13}+\frac{890329273}{2748235776}a^{12}+\frac{77170673}{85882368}a^{11}-\frac{175240319}{229019648}a^{10}-\frac{140385327}{114509824}a^{9}+\frac{95025803}{85882368}a^{8}+\frac{101001785}{42941184}a^{7}-\frac{88064293}{42941184}a^{6}-\frac{5430761}{1951872}a^{5}+\frac{332359}{121992}a^{4}+\frac{20626249}{5367648}a^{3}-\frac{9911705}{2683824}a^{2}-\frac{5394109}{670956}a+\frac{4903747}{670956}$, $\frac{65765}{229019648}a^{31}+\frac{2099961}{3664314368}a^{30}-\frac{94981}{161660928}a^{29}-\frac{63257}{53886976}a^{28}+\frac{2395}{1301248}a^{27}+\frac{1633117}{499679232}a^{26}-\frac{7898057}{1374117888}a^{25}-\frac{1290053}{114509824}a^{24}+\frac{8783639}{687058944}a^{23}+\frac{87031067}{3664314368}a^{22}-\frac{97007135}{2748235776}a^{21}-\frac{15485617}{229019648}a^{20}+\frac{42897449}{687058944}a^{19}+\frac{654467695}{5496471552}a^{18}-\frac{3072357}{19914752}a^{17}-\frac{794988373}{2748235776}a^{16}+\frac{51062191}{171764736}a^{15}+\frac{6469691083}{10992943104}a^{14}-\frac{1363852469}{2748235776}a^{13}-\frac{645002477}{687058944}a^{12}+\frac{738174353}{687058944}a^{11}+\frac{500496297}{229019648}a^{10}-\frac{264765563}{171764736}a^{9}-\frac{176561997}{57254912}a^{8}+\frac{41150553}{14313728}a^{7}+\frac{3562109}{622336}a^{6}-\frac{1570735}{487968}a^{5}-\frac{6744029}{975936}a^{4}+\frac{3062789}{670956}a^{3}+\frac{502140}{55913}a^{2}-\frac{5652371}{670956}a-\frac{12556937}{670956}$, $\frac{4675291}{10992943104}a^{31}+\frac{391315}{1832157184}a^{30}-\frac{4567567}{5496471552}a^{29}-\frac{1603633}{2748235776}a^{28}+\frac{1217837}{499679232}a^{27}+\frac{343999}{249839616}a^{26}-\frac{22543165}{2748235776}a^{25}-\frac{7448971}{1374117888}a^{24}+\frac{62768731}{3664314368}a^{23}+\frac{19437433}{1832157184}a^{22}-\frac{91953393}{1832157184}a^{21}-\frac{86159573}{2748235776}a^{20}+\frac{465454531}{5496471552}a^{19}+\frac{156626881}{2748235776}a^{18}-\frac{296603785}{1374117888}a^{17}-\frac{29067963}{229019648}a^{16}+\frac{4671110947}{10992943104}a^{15}+\frac{1584891953}{5496471552}a^{14}-\frac{1258213179}{1832157184}a^{13}-\frac{379705517}{916078592}a^{12}+\frac{733036195}{458039296}a^{11}+\frac{238793461}{229019648}a^{10}-\frac{16311475}{7468032}a^{9}-\frac{20493315}{14313728}a^{8}+\frac{367788397}{85882368}a^{7}+\frac{38392435}{14313728}a^{6}-\frac{2354605}{487968}a^{5}-\frac{438185}{121992}a^{4}+\frac{12417299}{1789216}a^{3}+\frac{10439753}{2683824}a^{2}-\frac{18290791}{1341912}a-\frac{2059933}{223652}$, $\frac{78421}{323321856}a^{31}-\frac{22171}{687058944}a^{30}-\frac{105097}{229019648}a^{29}+\frac{189523}{916078592}a^{28}+\frac{370351}{249839616}a^{27}-\frac{1119}{5204992}a^{26}-\frac{3302281}{687058944}a^{25}+\frac{2123461}{1374117888}a^{24}+\frac{20063301}{1832157184}a^{23}-\frac{2111947}{687058944}a^{22}-\frac{41102071}{1374117888}a^{21}+\frac{22263211}{2748235776}a^{20}+\frac{144208069}{2748235776}a^{19}-\frac{726769}{40415232}a^{18}-\frac{60997439}{458039296}a^{17}+\frac{40750949}{1374117888}a^{16}+\frac{1421307685}{5496471552}a^{15}-\frac{16083329}{171764736}a^{14}-\frac{26056483}{59744256}a^{13}+\frac{105671443}{916078592}a^{12}+\frac{224058591}{229019648}a^{11}-\frac{204840161}{687058944}a^{10}-\frac{238747559}{171764736}a^{9}+\frac{20873441}{42941184}a^{8}+\frac{55670045}{21470592}a^{7}-\frac{9798787}{14313728}a^{6}-\frac{496985}{162656}a^{5}+\frac{196587}{162656}a^{4}+\frac{751799}{167739}a^{3}-\frac{820865}{894608}a^{2}-\frac{1814951}{223652}a+\frac{828335}{223652}$, $\frac{4198735}{10992943104}a^{31}+\frac{245277}{1832157184}a^{30}-\frac{22419}{26943488}a^{29}-\frac{1927}{6735872}a^{28}+\frac{1108193}{499679232}a^{27}+\frac{168017}{249839616}a^{26}-\frac{3667113}{458039296}a^{25}-\frac{1860341}{687058944}a^{24}+\frac{62483519}{3664314368}a^{23}+\frac{31509301}{5496471552}a^{22}-\frac{130725989}{2748235776}a^{21}-\frac{21497875}{1374117888}a^{20}+\frac{476590883}{5496471552}a^{19}+\frac{84353683}{2748235776}a^{18}-\frac{565086031}{2748235776}a^{17}-\frac{32483801}{458039296}a^{16}+\frac{1554632149}{3664314368}a^{15}+\frac{802968775}{5496471552}a^{14}-\frac{1863406615}{2748235776}a^{13}-\frac{113181803}{458039296}a^{12}+\frac{1058490427}{687058944}a^{11}+\frac{88709455}{171764736}a^{10}-\frac{382053799}{171764736}a^{9}-\frac{72186611}{85882368}a^{8}+\frac{21915625}{5367648}a^{7}+\frac{30591269}{21470592}a^{6}-\frac{4893337}{975936}a^{5}-\frac{208723}{121992}a^{4}+\frac{8992169}{1341912}a^{3}+\frac{310357}{111826}a^{2}-\frac{9062401}{670956}a-\frac{778253}{167739}$, $\frac{10503403}{7328628736}a^{31}+\frac{8014257}{3664314368}a^{30}-\frac{15384107}{5496471552}a^{29}-\frac{1490371}{343529472}a^{28}+\frac{8624359}{999358464}a^{27}+\frac{284579}{21725184}a^{26}-\frac{39098707}{1374117888}a^{25}-\frac{60096397}{1374117888}a^{24}+\frac{1341707123}{21985886208}a^{23}+\frac{343886755}{3664314368}a^{22}-\frac{477781945}{2748235776}a^{21}-\frac{736218203}{2748235776}a^{20}+\frac{1106230863}{3664314368}a^{19}+\frac{2560123855}{5496471552}a^{18}-\frac{1384984993}{1832157184}a^{17}-\frac{62461765}{53886976}a^{16}+\frac{32843825473}{21985886208}a^{15}+\frac{8455620161}{3664314368}a^{14}-\frac{2235162169}{916078592}a^{13}-\frac{3433490771}{916078592}a^{12}+\frac{7627699405}{1374117888}a^{11}+\frac{1472553499}{171764736}a^{10}-\frac{890150805}{114509824}a^{9}-\frac{686789823}{57254912}a^{8}+\frac{1254942259}{85882368}a^{7}+\frac{161117757}{7156864}a^{6}-\frac{11182895}{650624}a^{5}-\frac{1132199}{42432}a^{4}+\frac{460565}{19448}a^{3}+\frac{97100119}{2683824}a^{2}-\frac{61133183}{1341912}a-\frac{11977375}{167739}$, $\frac{62013}{229019648}a^{31}-\frac{3276073}{3664314368}a^{30}-\frac{6199}{13471744}a^{29}+\frac{309643}{161660928}a^{28}+\frac{100859}{62459904}a^{27}-\frac{2708509}{499679232}a^{26}-\frac{1741159}{343529472}a^{25}+\frac{6320891}{343529472}a^{24}+\frac{7681163}{687058944}a^{23}-\frac{433605313}{10992943104}a^{22}-\frac{7105481}{229019648}a^{21}+\frac{76969207}{687058944}a^{20}+\frac{36118691}{687058944}a^{19}-\frac{1077203471}{5496471552}a^{18}-\frac{47714651}{343529472}a^{17}+\frac{1321400453}{2748235776}a^{16}+\frac{58779075}{229019648}a^{15}-\frac{10727625499}{10992943104}a^{14}-\frac{301436125}{687058944}a^{13}+\frac{357745035}{229019648}a^{12}+\frac{223119653}{229019648}a^{11}-\frac{2457076409}{687058944}a^{10}-\frac{38855015}{28627456}a^{9}+\frac{868268977}{171764736}a^{8}+\frac{18525031}{7156864}a^{7}-\frac{17439377}{1867008}a^{6}-\frac{113193}{40664}a^{5}+\frac{11004625}{975936}a^{4}+\frac{12048947}{2683824}a^{3}-\frac{822400}{55913}a^{2}-\frac{4631183}{670956}a+\frac{20206957}{670956}$, $\frac{2168473}{21985886208}a^{31}-\frac{25057}{5496471552}a^{30}-\frac{1376621}{5496471552}a^{29}-\frac{60179}{687058944}a^{28}+\frac{591391}{999358464}a^{27}-\frac{4565}{83279872}a^{26}-\frac{254307}{114509824}a^{25}-\frac{105957}{229019648}a^{24}+\frac{32546729}{7328628736}a^{23}+\frac{929615}{1832157184}a^{22}-\frac{1499237}{114509824}a^{21}-\frac{811249}{1374117888}a^{20}+\frac{80904023}{3664314368}a^{19}+\frac{6118209}{916078592}a^{18}-\frac{295730191}{5496471552}a^{17}-\frac{2179399}{1374117888}a^{16}+\frac{2590844921}{21985886208}a^{15}+\frac{48453957}{1832157184}a^{14}-\frac{29217971}{171764736}a^{13}-\frac{33148667}{1374117888}a^{12}+\frac{198163483}{458039296}a^{11}+\frac{884551}{14936064}a^{10}-\frac{11733133}{20207616}a^{9}-\frac{4004199}{28627456}a^{8}+\frac{95693123}{85882368}a^{7}+\frac{127723}{1789216}a^{6}-\frac{2948563}{1951872}a^{5}-\frac{26731}{60996}a^{4}+\frac{3571}{2431}a^{3}-\frac{222553}{1341912}a^{2}-\frac{1971241}{447304}a-\frac{65812}{167739}$, $\frac{650525}{7328628736}a^{31}+\frac{11567}{38436864}a^{30}-\frac{486169}{2748235776}a^{29}-\frac{4569}{6406144}a^{28}+\frac{162011}{333119488}a^{27}+\frac{34795}{19218432}a^{26}-\frac{220205}{119488512}a^{25}-\frac{20651}{3203072}a^{24}+\frac{80639093}{21985886208}a^{23}+\frac{175951}{12812288}a^{22}-\frac{21235119}{1832157184}a^{21}-\frac{10787}{278528}a^{20}+\frac{231997123}{10992943104}a^{19}+\frac{1362623}{19218432}a^{18}-\frac{274208471}{5496471552}a^{17}-\frac{809881}{4804608}a^{16}+\frac{785955597}{7328628736}a^{15}+\frac{13540727}{38436864}a^{14}-\frac{878626175}{5496471552}a^{13}-\frac{10726277}{19218432}a^{12}+\frac{8590421}{21470592}a^{11}+\frac{2063961}{1601536}a^{10}-\frac{63610095}{114509824}a^{9}-\frac{92787}{50048}a^{8}+\frac{40631491}{42941184}a^{7}+\frac{338545}{100096}a^{6}-\frac{2788141}{1951872}a^{5}-\frac{159245}{37536}a^{4}+\frac{2833901}{1789216}a^{3}+\frac{31633}{6256}a^{2}-\frac{2565223}{670956}a-\frac{3233}{276}$, $\frac{3432295}{7328628736}a^{31}-\frac{32843}{38436864}a^{30}-\frac{2897951}{2748235776}a^{29}+\frac{12209}{6406144}a^{28}+\frac{2821859}{999358464}a^{27}-\frac{32437}{6406144}a^{26}-\frac{1212847}{119488512}a^{25}+\frac{175313}{9609216}a^{24}+\frac{153792037}{7328628736}a^{23}-\frac{478987}{12812288}a^{22}-\frac{109970885}{1832157184}a^{21}+\frac{90713}{835584}a^{20}+\frac{1185219929}{10992943104}a^{19}-\frac{3711619}{19218432}a^{18}-\frac{467838103}{1832157184}a^{17}+\frac{738297}{1601536}a^{16}+\frac{3900556087}{7328628736}a^{15}-\frac{12361489}{12812288}a^{14}-\frac{1515652519}{1832157184}a^{13}+\frac{28431253}{19218432}a^{12}+\frac{671193379}{343529472}a^{11}-\frac{16950325}{4804608}a^{10}-\frac{313111001}{114509824}a^{9}+\frac{983651}{200192}a^{8}+\frac{4434995}{894608}a^{7}-\frac{2714863}{300288}a^{6}-\frac{12384095}{1951872}a^{5}+\frac{141583}{12512}a^{4}+\frac{40874081}{5367648}a^{3}-\frac{86737}{6256}a^{2}-\frac{2763647}{167739}a+\frac{8459}{276}$, $\frac{5370821}{21985886208}a^{31}-\frac{1355373}{3664314368}a^{30}-\frac{152731}{343529472}a^{29}+\frac{1083947}{1374117888}a^{28}+\frac{473729}{333119488}a^{27}-\frac{80689}{29392896}a^{26}-\frac{4063073}{916078592}a^{25}+\frac{633055}{80830464}a^{24}+\frac{206376223}{21985886208}a^{23}-\frac{194106437}{10992943104}a^{22}-\frac{144826325}{5496471552}a^{21}+\frac{7860761}{161660928}a^{20}+\frac{498671209}{10992943104}a^{19}-\frac{471901363}{5496471552}a^{18}-\frac{634141705}{5496471552}a^{17}+\frac{590622575}{2748235776}a^{16}+\frac{4770879253}{21985886208}a^{15}-\frac{1492613533}{3664314368}a^{14}-\frac{114407335}{323321856}a^{13}+\frac{645041105}{916078592}a^{12}+\frac{90714147}{114509824}a^{11}-\frac{127113263}{85882368}a^{10}-\frac{124908701}{114509824}a^{9}+\frac{370262701}{171764736}a^{8}+\frac{45038111}{21470592}a^{7}-\frac{14616881}{3578432}a^{6}-\frac{4271653}{1951872}a^{5}+\frac{1490049}{325312}a^{4}+\frac{5251281}{1789216}a^{3}-\frac{17385113}{2683824}a^{2}-\frac{1918403}{335478}a+\frac{3395335}{335478}$, $\frac{207693}{916078592}a^{31}-\frac{336823}{999358464}a^{30}-\frac{3019495}{5496471552}a^{29}+\frac{5065}{7807488}a^{28}+\frac{59467}{41639936}a^{27}-\frac{1103051}{499679232}a^{26}-\frac{13585555}{2748235776}a^{25}+\frac{858577}{124919808}a^{24}+\frac{28706537}{2748235776}a^{23}-\frac{14613397}{999358464}a^{22}-\frac{55149023}{1832157184}a^{21}+\frac{3608541}{83279872}a^{20}+\frac{37358149}{687058944}a^{19}-\frac{12128097}{166559744}a^{18}-\frac{114499269}{916078592}a^{17}+\frac{46855171}{249839616}a^{16}+\frac{745327421}{2748235776}a^{15}-\frac{359089063}{999358464}a^{14}-\frac{2242454783}{5496471552}a^{13}+\frac{153697421}{249839616}a^{12}+\frac{440618031}{458039296}a^{11}-\frac{103267}{76544}a^{10}-\frac{228823865}{171764736}a^{9}+\frac{29075489}{15614976}a^{8}+\frac{212753527}{85882368}a^{7}-\frac{417407}{114816}a^{6}-\frac{2990137}{975936}a^{5}+\frac{229007}{57408}a^{4}+\frac{18519673}{5367648}a^{3}-\frac{1370011}{243984}a^{2}-\frac{11887285}{1341912}a+\frac{52103}{5083}$ 598823/5496471552a^30 - 351331/1374117888a^28 + 152897/249839616a^26 - 112259/42941184a^24 + 28356293/5496471552a^22 - 656639/42941184a^20 + 25178825/916078592a^18 - 87450241/1374117888a^16 + 252322157/1832157184a^14 - 17461369/85882368a^12 + 58833033/114509824a^10 - 15680039/21470592a^8 + 9091837/7156864a^6 - 421673/243984a^4 + 127086/55913a^2 - 18737/3289, 1215085/10992943104a^31 - 1394149/5496471552a^29 + 309323/499679232a^27 - 97963/39829504a^25 + 18384557/3664314368a^23 - 26777331/1832157184a^21 + 8784421/323321856a^19 - 82237741/1374117888a^17 + 1497652933/10992943104a^15 - 1103988403/5496471552a^13 + 220163729/458039296a^11 - 118625219/171764736a^9 + 104765123/85882368a^7 - 1638191/975936a^5 + 2897783/1789216a^3 - 5762987/1341912a - 1, 1468835/7328628736a^31 - 689955/1832157184a^29 + 1209311/999358464a^27 - 116087/29872128a^25 + 186816811/21985886208a^23 - 11159727/458039296a^21 + 9009127/215547904a^19 - 190415389/1832157184a^17 + 4584444361/21985886208a^15 - 473411611/1374117888a^13 + 1066638187/1374117888a^11 - 376994399/343529472a^9 + 58311373/28627456a^7 - 4761553/1951872a^5 + 2879671/894608a^3 - 8573359/1341912a - 1, 5280841/21985886208a^31 - 875341/5496471552a^30 - 589265/1374117888a^29 + 287897/916078592a^28 + 1334431/999358464a^27 - 219595/249839616a^26 - 12673747/2748235776a^25 + 4626629/1374117888a^24 + 71284601/7328628736a^23 - 2384983/323321856a^22 - 51928967/1832157184a^21 + 58607275/2748235776a^20 + 176210079/3664314368a^19 - 34565103/916078592a^18 - 221641559/1832157184a^17 + 16259467/171764736a^16 + 5212804121/21985886208a^15 - 1107454741/5496471552a^14 - 90750593/238977024a^13 + 890329273/2748235776a^12 + 77170673/85882368a^11 - 175240319/229019648a^10 - 140385327/114509824a^9 + 95025803/85882368a^8 + 101001785/42941184a^7 - 88064293/42941184a^6 - 5430761/1951872a^5 + 332359/121992a^4 + 20626249/5367648a^3 - 9911705/2683824a^2 - 5394109/670956a + 4903747/670956, 65765/229019648a^31 + 2099961/3664314368a^30 - 94981/161660928a^29 - 63257/53886976a^28 + 2395/1301248a^27 + 1633117/499679232a^26 - 7898057/1374117888a^25 - 1290053/114509824a^24 + 8783639/687058944a^23 + 87031067/3664314368a^22 - 97007135/2748235776a^21 - 15485617/229019648a^20 + 42897449/687058944a^19 + 654467695/5496471552a^18 - 3072357/19914752a^17 - 794988373/2748235776a^16 + 51062191/171764736a^15 + 6469691083/10992943104a^14 - 1363852469/2748235776a^13 - 645002477/687058944a^12 + 738174353/687058944a^11 + 500496297/229019648a^10 - 264765563/171764736a^9 - 176561997/57254912a^8 + 41150553/14313728a^7 + 3562109/622336a^6 - 1570735/487968a^5 - 6744029/975936a^4 + 3062789/670956a^3 + 502140/55913a^2 - 5652371/670956a - 12556937/670956, 4675291/10992943104a^31 + 391315/1832157184a^30 - 4567567/5496471552a^29 - 1603633/2748235776a^28 + 1217837/499679232a^27 + 343999/249839616a^26 - 22543165/2748235776a^25 - 7448971/1374117888a^24 + 62768731/3664314368a^23 + 19437433/1832157184a^22 - 91953393/1832157184a^21 - 86159573/2748235776a^20 + 465454531/5496471552a^19 + 156626881/2748235776a^18 - 296603785/1374117888a^17 - 29067963/229019648a^16 + 4671110947/10992943104a^15 + 1584891953/5496471552a^14 - 1258213179/1832157184a^13 - 379705517/916078592a^12 + 733036195/458039296a^11 + 238793461/229019648a^10 - 16311475/7468032a^9 - 20493315/14313728a^8 + 367788397/85882368a^7 + 38392435/14313728a^6 - 2354605/487968a^5 - 438185/121992a^4 + 12417299/1789216a^3 + 10439753/2683824a^2 - 18290791/1341912a - 2059933/223652, 78421/323321856a^31 - 22171/687058944a^30 - 105097/229019648a^29 + 189523/916078592a^28 + 370351/249839616a^27 - 1119/5204992a^26 - 3302281/687058944a^25 + 2123461/1374117888a^24 + 20063301/1832157184a^23 - 2111947/687058944a^22 - 41102071/1374117888a^21 + 22263211/2748235776a^20 + 144208069/2748235776a^19 - 726769/40415232a^18 - 60997439/458039296a^17 + 40750949/1374117888a^16 + 1421307685/5496471552a^15 - 16083329/171764736a^14 - 26056483/59744256a^13 + 105671443/916078592a^12 + 224058591/229019648a^11 - 204840161/687058944a^10 - 238747559/171764736a^9 + 20873441/42941184a^8 + 55670045/21470592a^7 - 9798787/14313728a^6 - 496985/162656a^5 + 196587/162656a^4 + 751799/167739a^3 - 820865/894608a^2 - 1814951/223652a + 828335/223652, 4198735/10992943104a^31 + 245277/1832157184a^30 - 22419/26943488a^29 - 1927/6735872a^28 + 1108193/499679232a^27 + 168017/249839616a^26 - 3667113/458039296a^25 - 1860341/687058944a^24 + 62483519/3664314368a^23 + 31509301/5496471552a^22 - 130725989/2748235776a^21 - 21497875/1374117888a^20 + 476590883/5496471552a^19 + 84353683/2748235776a^18 - 565086031/2748235776a^17 - 32483801/458039296a^16 + 1554632149/3664314368a^15 + 802968775/5496471552a^14 - 1863406615/2748235776a^13 - 113181803/458039296a^12 + 1058490427/687058944a^11 + 88709455/171764736a^10 - 382053799/171764736a^9 - 72186611/85882368a^8 + 21915625/5367648a^7 + 30591269/21470592a^6 - 4893337/975936a^5 - 208723/121992a^4 + 8992169/1341912a^3 + 310357/111826a^2 - 9062401/670956a - 778253/167739, 10503403/7328628736a^31 + 8014257/3664314368a^30 - 15384107/5496471552a^29 - 1490371/343529472a^28 + 8624359/999358464a^27 + 284579/21725184a^26 - 39098707/1374117888a^25 - 60096397/1374117888a^24 + 1341707123/21985886208a^23 + 343886755/3664314368a^22 - 477781945/2748235776a^21 - 736218203/2748235776a^20 + 1106230863/3664314368a^19 + 2560123855/5496471552a^18 - 1384984993/1832157184a^17 - 62461765/53886976a^16 + 32843825473/21985886208a^15 + 8455620161/3664314368a^14 - 2235162169/916078592a^13 - 3433490771/916078592a^12 + 7627699405/1374117888a^11 + 1472553499/171764736a^10 - 890150805/114509824a^9 - 686789823/57254912a^8 + 1254942259/85882368a^7 + 161117757/7156864a^6 - 11182895/650624a^5 - 1132199/42432a^4 + 460565/19448a^3 + 97100119/2683824a^2 - 61133183/1341912a - 11977375/167739, 62013/229019648a^31 - 3276073/3664314368a^30 - 6199/13471744a^29 + 309643/161660928a^28 + 100859/62459904a^27 - 2708509/499679232a^26 - 1741159/343529472a^25 + 6320891/343529472a^24 + 7681163/687058944a^23 - 433605313/10992943104a^22 - 7105481/229019648a^21 + 76969207/687058944a^20 + 36118691/687058944a^19 - 1077203471/5496471552a^18 - 47714651/343529472a^17 + 1321400453/2748235776a^16 + 58779075/229019648a^15 - 10727625499/10992943104a^14 - 301436125/687058944a^13 + 357745035/229019648a^12 + 223119653/229019648a^11 - 2457076409/687058944a^10 - 38855015/28627456a^9 + 868268977/171764736a^8 + 18525031/7156864a^7 - 17439377/1867008a^6 - 113193/40664a^5 + 11004625/975936a^4 + 12048947/2683824a^3 - 822400/55913a^2 - 4631183/670956a + 20206957/670956, 2168473/21985886208a^31 - 25057/5496471552a^30 - 1376621/5496471552a^29 - 60179/687058944a^28 + 591391/999358464a^27 - 4565/83279872a^26 - 254307/114509824a^25 - 105957/229019648a^24 + 32546729/7328628736a^23 + 929615/1832157184a^22 - 1499237/114509824a^21 - 811249/1374117888a^20 + 80904023/3664314368a^19 + 6118209/916078592a^18 - 295730191/5496471552a^17 - 2179399/1374117888a^16 + 2590844921/21985886208a^15 + 48453957/1832157184a^14 - 29217971/171764736a^13 - 33148667/1374117888a^12 + 198163483/458039296a^11 + 884551/14936064a^10 - 11733133/20207616a^9 - 4004199/28627456a^8 + 95693123/85882368a^7 + 127723/1789216a^6 - 2948563/1951872a^5 - 26731/60996a^4 + 3571/2431a^3 - 222553/1341912a^2 - 1971241/447304a - 65812/167739, 650525/7328628736a^31 + 11567/38436864a^30 - 486169/2748235776a^29 - 4569/6406144a^28 + 162011/333119488a^27 + 34795/19218432a^26 - 220205/119488512a^25 - 20651/3203072a^24 + 80639093/21985886208a^23 + 175951/12812288a^22 - 21235119/1832157184a^21 - 10787/278528a^20 + 231997123/10992943104a^19 + 1362623/19218432a^18 - 274208471/5496471552a^17 - 809881/4804608a^16 + 785955597/7328628736a^15 + 13540727/38436864a^14 - 878626175/5496471552a^13 - 10726277/19218432a^12 + 8590421/21470592a^11 + 2063961/1601536a^10 - 63610095/114509824a^9 - 92787/50048a^8 + 40631491/42941184a^7 + 338545/100096a^6 - 2788141/1951872a^5 - 159245/37536a^4 + 2833901/1789216a^3 + 31633/6256a^2 - 2565223/670956a - 3233/276, 3432295/7328628736a^31 - 32843/38436864a^30 - 2897951/2748235776a^29 + 12209/6406144a^28 + 2821859/999358464a^27 - 32437/6406144a^26 - 1212847/119488512a^25 + 175313/9609216a^24 + 153792037/7328628736a^23 - 478987/12812288a^22 - 109970885/1832157184a^21 + 90713/835584a^20 + 1185219929/10992943104a^19 - 3711619/19218432a^18 - 467838103/1832157184a^17 + 738297/1601536a^16 + 3900556087/7328628736a^15 - 12361489/12812288a^14 - 1515652519/1832157184a^13 + 28431253/19218432a^12 + 671193379/343529472a^11 - 16950325/4804608a^10 - 313111001/114509824a^9 + 983651/200192a^8 + 4434995/894608a^7 - 2714863/300288a^6 - 12384095/1951872a^5 + 141583/12512a^4 + 40874081/5367648a^3 - 86737/6256a^2 - 2763647/167739a + 8459/276, 5370821/21985886208a^31 - 1355373/3664314368a^30 - 152731/343529472a^29 + 1083947/1374117888a^28 + 473729/333119488a^27 - 80689/29392896a^26 - 4063073/916078592a^25 + 633055/80830464a^24 + 206376223/21985886208a^23 - 194106437/10992943104a^22 - 144826325/5496471552a^21 + 7860761/161660928a^20 + 498671209/10992943104a^19 - 471901363/5496471552a^18 - 634141705/5496471552a^17 + 590622575/2748235776a^16 + 4770879253/21985886208a^15 - 1492613533/3664314368a^14 - 114407335/323321856a^13 + 645041105/916078592a^12 + 90714147/114509824a^11 - 127113263/85882368a^10 - 124908701/114509824a^9 + 370262701/171764736a^8 + 45038111/21470592a^7 - 14616881/3578432a^6 - 4271653/1951872a^5 + 1490049/325312a^4 + 5251281/1789216a^3 - 17385113/2683824a^2 - 1918403/335478a + 3395335/335478, 207693/916078592a^31 - 336823/999358464a^30 - 3019495/5496471552a^29 + 5065/7807488a^28 + 59467/41639936a^27 - 1103051/499679232a^26 - 13585555/2748235776a^25 + 858577/124919808a^24 + 28706537/2748235776a^23 - 14613397/999358464a^22 - 55149023/1832157184a^21 + 3608541/83279872a^20 + 37358149/687058944a^19 - 12128097/166559744a^18 - 114499269/916078592a^17 + 46855171/249839616a^16 + 745327421/2748235776a^15 - 359089063/999358464a^14 - 2242454783/5496471552a^13 + 153697421/249839616a^12 + 440618031/458039296a^11 - 103267/76544a^10 - 228823865/171764736a^9 + 29075489/15614976a^8 + 212753527/85882368a^7 - 417407/114816a^6 - 2990137/975936a^5 + 229007/57408a^4 + 18519673/5367648a^3 - 1370011/243984a^2 - 11887285/1341912*a + 52103/5083 (assuming GRH)	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:	$ 32664023933.600838 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 32664023933.600838 \cdot 4}{24\cdot\sqrt{14680252274932068764306928624388060814210236416}}\cr\approx \mathstrut & 0.265112111724684 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536)

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^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536, 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^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536);

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^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536);

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

$D_4^2:C_2^3$ (as 32T12882):

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 solvable group of order 512

The 80 conjugacy class representatives for $D_4^2:C_2^3$ are not computed

Character table for $D_4^2:C_2^3$ is not computed

Intermediate fields

$\Q(\sqrt{-1}) $, $\Q(\sqrt{6}) $, $\Q(\sqrt{-2}) $, $\Q(\sqrt{3}) $, $\Q(\sqrt{-6}) $, $\Q(\sqrt{2}) $, $\Q(\sqrt{-3}) $, 4.0.29952.1, 4.4.7488.1, 4.0.7488.1, 4.4.29952.1, $\Q(\sqrt{2}, \sqrt{-3})$, $\Q(i, \sqrt{6})$, $\Q(\sqrt{-2}, \sqrt{3})$, $\Q(\zeta_{8})$, $\Q(\sqrt{2}, \sqrt{3})$, $\Q(\zeta_{12})$, $\Q(\sqrt{-2}, \sqrt{-3})$, 8.8.348083453952.1, 8.0.5438803968.1, 8.8.5438803968.1, 8.0.348083453952.1, $\Q(\zeta_{24})$, 8.0.3588489216.16, 8.0.3588489216.5, 8.0.3588489216.11, 8.8.3588489216.1, 8.0.897122304.10, 8.0.56070144.2, 16.0.12877254853348294656.1, 16.0.121162090915154104418304.2, 16.0.121162090915154104418304.3, 16.16.121162090915154104418304.1, 16.0.121162090915154104418304.1, 16.0.29580588602332545024.3, 16.0.121162090915154104418304.4

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]

Sibling fields

Degree 32 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.8.0.1}{8} }^{4}$	${\href{/padicField/7.8.0.1}{8} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$	R	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$	${\href{/padicField/19.8.0.1}{8} }^{4}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$	${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{8}$	${\href{/padicField/31.4.0.1}{4} }^{8}$	${\href{/padicField/37.2.0.1}{2} }^{16}$	${\href{/padicField/41.8.0.1}{8} }^{4}$	${\href{/padicField/43.4.0.1}{4} }^{8}$	${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$	${\href{/padicField/53.2.0.1}{2} }^{16}$	${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$

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
$2$ 2	2.16.36.1	$x^{16} + 8 x^{12} + 8 x^{11} + 24 x^{10} + 44 x^{8} - 16 x^{7} + 16 x^{6} - 56 x^{4} + 112 x^{3} + 196$	$8$	$2$	$36$	$D_4\times C_2$	$[2, 2, 3]^{2}$
$2$ 2	2.16.36.1	$x^{16} + 8 x^{12} + 8 x^{11} + 24 x^{10} + 44 x^{8} - 16 x^{7} + 16 x^{6} - 56 x^{4} + 112 x^{3} + 196$	$8$	$2$	$36$	$D_4\times C_2$	$[2, 2, 3]^{2}$
$3$ 3	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$13$ 13	13.2.0.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.2.0.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.2.0.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.2.0.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.2.0.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.2.0.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.2.0.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.2.0.1	$x^{2} + 12 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$97$ 97	97.2.1.1	$x^{2} + 97$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	97.2.1.1	$x^{2} + 97$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	97.2.0.1	$x^{2} + 96 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	97.2.0.1	$x^{2} + 96 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	97.2.0.1	$x^{2} + 96 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	97.2.0.1	$x^{2} + 96 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	97.2.1.1	$x^{2} + 97$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	97.2.1.1	$x^{2} + 97$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	97.4.0.1	$x^{4} + 6 x^{2} + 80 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	97.4.0.1	$x^{4} + 6 x^{2} + 80 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	97.4.0.1	$x^{4} + 6 x^{2} + 80 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	97.4.0.1	$x^{4} + 6 x^{2} + 80 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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