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Note: Search results may be incomplete. Given $p$-adic completions contain an unramified field and completions are only searched for ramified primes.

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Degree Signature Ramified prime count Ramified Unramified primes
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Regulator $p$-adic completions Monogenic Index Ines. \(p\)
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Results (1-50 of 541219 matches)

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Label Polynomial Discriminant Galois group Class group Regulator
4.0.117.1 $x^{4} - x^{3} - x^{2} + x + 1$ $3^{2}\cdot 13$ $D_{4}$ (as 4T3) trivial $0.543535072498$
4.0.832.1 $x^{4} - 2 x^{3} - 2 x + 5$ $2^{6}\cdot 13$ $D_{4}$ (as 4T3) trivial $2.13862208632$
4.0.1040.2 $x^{4} + x^{2} - 4 x + 4$ $2^{4}\cdot 5\cdot 13$ $D_{4}$ (as 4T3) trivial $3.46864904298$
4.2.1456.1 $x^{4} - 2 x^{2} - 2 x + 1$ $-\,2^{4}\cdot 7\cdot 13$ $S_4$ (as 4T5) trivial $1.76052319498$
4.0.1872.1 $x^{4} + 7 x^{2} + 13$ $2^{4}\cdot 3^{2}\cdot 13$ $D_{4}$ (as 4T3) $[2]$ $2.575278132$
4.2.1963.1 $x^{4} - x^{3} + 2 x^{2} - 2 x - 1$ $-\,13\cdot 151$ $S_4$ (as 4T5) trivial $1.47885191076$
4.2.2327.1 $x^{4} - x^{2} - x - 2$ $-\,13\cdot 179$ $S_4$ (as 4T5) trivial $2.69444673434$
4.0.2457.1 $x^{4} - x^{3} + 9 x^{2} - 4 x + 16$ $3^{3}\cdot 7\cdot 13$ $D_{4}$ (as 4T3) $[2]$ $2.87383998018$
4.0.3757.1 $x^{4} - 2 x^{3} + 6 x^{2} - 5 x + 2$ $13\cdot 17^{2}$ $D_{4}$ (as 4T3) trivial $4.18942509452$
4.0.4212.1 $x^{4} - x^{3} + 3 x^{2} + x + 2$ $2^{2}\cdot 3^{4}\cdot 13$ $S_4$ (as 4T5) trivial $5.85087884431$
4.0.4329.2 $x^{4} - x^{3} + 11 x^{2} - 2 x + 28$ $3^{2}\cdot 13\cdot 37$ $D_{4}$ (as 4T3) $[2]$ $3.01636186336$
4.0.5200.1 $x^{4} + 4 x^{2} - 2 x + 2$ $2^{4}\cdot 5^{2}\cdot 13$ $S_4$ (as 4T5) trivial $3.99769759669$
4.2.5343.1 $x^{4} - 2 x^{3} - x^{2} + x - 2$ $-\,3\cdot 13\cdot 137$ $S_4$ (as 4T5) trivial $4.59806787858$
4.2.5616.1 $x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 2$ $-\,2^{4}\cdot 3^{3}\cdot 13$ $D_{4}$ (as 4T3) trivial $7.10823929493$
4.2.5616.2 $x^{4} - 3 x^{2} - 6 x - 3$ $-\,2^{4}\cdot 3^{3}\cdot 13$ $D_{4}$ (as 4T3) trivial $3.25525922497$
4.2.6071.1 $x^{4} - x^{3} + 3 x^{2} + 2 x - 1$ $-\,13\cdot 467$ $S_4$ (as 4T5) trivial $3.98583634371$
4.2.6604.1 $x^{4} - x^{3} - 2 x^{2} - 2 x + 2$ $-\,2^{2}\cdot 13\cdot 127$ $S_4$ (as 4T5) trivial $6.73517393845$
4.2.6656.1 $x^{4} - 4 x + 1$ $-\,2^{9}\cdot 13$ $S_4$ (as 4T5) trivial $5.60630929506$
4.0.6656.2 $x^{4} - 2 x^{2} + 26$ $2^{9}\cdot 13$ $D_{4}$ (as 4T3) trivial $8.57574318427$
4.0.6669.1 $x^{4} - x^{3} + 3$ $3^{3}\cdot 13\cdot 19$ $S_4$ (as 4T5) trivial $3.1153669263$
4.0.6669.2 $x^{4} - 2 x^{3} - 12 x^{2} + 13 x + 43$ $3^{3}\cdot 13\cdot 19$ $D_{4}$ (as 4T3) $[2]$ $3.26087355202$
4.0.7488.4 $x^{4} - 2 x^{3} - 4 x^{2} - 4 x + 34$ $2^{6}\cdot 3^{2}\cdot 13$ $D_{4}$ (as 4T3) $[2]$ $3.72832308832$
4.0.7540.1 $x^{4} - x^{3} + 5 x^{2} - x + 6$ $2^{2}\cdot 5\cdot 13\cdot 29$ $S_4$ (as 4T5) trivial $7.58827081214$
4.0.7696.1 $x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 29$ $2^{4}\cdot 13\cdot 37$ $D_{4}$ (as 4T3) trivial $6.89361833182$
4.0.7709.1 $x^{4} - 2 x^{3} + 6 x^{2} - 3 x + 2$ $13\cdot 593$ $S_4$ (as 4T5) trivial $6.99651976044$
4.2.8203.1 $x^{4} - 4 x^{2} - 3 x + 2$ $-\,13\cdot 631$ $S_4$ (as 4T5) trivial $8.52957793619$
4.0.8541.2 $x^{4} - 2 x^{3} + 16 x^{2} - 15 x + 63$ $3^{2}\cdot 13\cdot 73$ $D_{4}$ (as 4T3) $[2]$ $5.99080303274$
4.2.8684.1 $x^{4} - 6 x + 1$ $-\,2^{2}\cdot 13\cdot 167$ $S_4$ (as 4T5) trivial $8.43851953642$
4.0.9217.1 $x^{4} - x^{3} + 2 x^{2} - 3 x + 4$ $13\cdot 709$ $S_4$ (as 4T5) trivial $5.17970176411$
4.0.9360.2 $x^{4} - x^{2} - 12 x + 37$ $2^{4}\cdot 3^{2}\cdot 5\cdot 13$ $D_{4}$ (as 4T3) $[2]$ $3.20082011047$
4.2.10192.1 $x^{4} - 2 x^{3} - x^{2} - 4 x - 1$ $-\,2^{4}\cdot 7^{2}\cdot 13$ $S_4$ (as 4T5) trivial $4.66305911124$
4.2.10296.1 $x^{4} - x^{2} - 6 x + 4$ $-\,2^{3}\cdot 3^{2}\cdot 11\cdot 13$ $S_4$ (as 4T5) trivial $6.83369049492$
4.2.10348.1 $x^{4} - x^{3} - 2 x^{2} - 4 x + 4$ $-\,2^{2}\cdot 13\cdot 199$ $S_4$ (as 4T5) trivial $9.07588102802$
4.0.10712.1 $x^{4} - x^{3} - 2 x^{2} + x + 11$ $2^{3}\cdot 13\cdot 103$ $S_4$ (as 4T5) trivial $5.65602335603$
4.2.10751.1 $x^{4} - x^{3} + 2 x^{2} - 3 x - 2$ $-\,13\cdot 827$ $S_4$ (as 4T5) trivial $7.69848527608$
4.0.10881.2 $x^{4} - x^{3} - 15 x^{2} + 14 x + 76$ $3^{3}\cdot 13\cdot 31$ $D_{4}$ (as 4T3) $[2]$ $6.32106253222$
4.0.11024.1 $x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 40$ $2^{4}\cdot 13\cdot 53$ $D_{4}$ (as 4T3) trivial $18.8280785288$
4.0.11024.2 $x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 41$ $2^{4}\cdot 13\cdot 53$ $D_{4}$ (as 4T3) trivial $3.31738659771$
4.0.11232.1 $x^{4} - 3 x^{2} - 2 x + 6$ $2^{5}\cdot 3^{3}\cdot 13$ $S_4$ (as 4T5) trivial $11.2180943751$
4.0.11349.1 $x^{4} - x^{3} - 17 x^{2} + 9 x + 81$ $3^{2}\cdot 13\cdot 97$ $D_{4}$ (as 4T3) $[2]$ $4.96294344567$
4.0.11505.1 $x^{4} - 2 x^{3} + 5 x^{2} - 3 x + 6$ $3\cdot 5\cdot 13\cdot 59$ $S_4$ (as 4T5) trivial $6.18991850637$
4.2.11843.1 $x^{4} - x^{3} + 4 x^{2} + 2 x - 1$ $-\,13\cdot 911$ $S_4$ (as 4T5) trivial $5.18761895339$
4.2.12103.1 $x^{4} - x^{3} + 7 x^{2} - 6 x + 1$ $-\,7^{2}\cdot 13\cdot 19$ $S_4$ (as 4T5) trivial $6.77715534445$
4.0.12688.1 $x^{4} - 13 x^{2} - 4 x + 53$ $2^{4}\cdot 13\cdot 61$ $D_{4}$ (as 4T3) trivial $6.80302010162$
4.0.12688.2 $x^{4} - x^{2} - 14 x + 50$ $2^{4}\cdot 13\cdot 61$ $D_{4}$ (as 4T3) trivial $17.8529648212$
4.0.12753.2 $x^{4} - 2 x^{3} + 19 x^{2} - 18 x + 93$ $3^{2}\cdot 13\cdot 109$ $D_{4}$ (as 4T3) $[2]$ $9.30491271006$
4.0.13104.1 $x^{4} - 19 x^{2} + 91$ $2^{4}\cdot 3^{2}\cdot 7\cdot 13$ $D_{4}$ (as 4T3) $[2]$ $3.61231679274$
4.0.13104.2 $x^{4} + 19 x^{2} + 91$ $2^{4}\cdot 3^{2}\cdot 7\cdot 13$ $D_{4}$ (as 4T3) $[2]$ $8.40429713417$
4.0.13273.1 $x^{4} - x^{3} + x^{2} + 4 x + 3$ $13\cdot 1021$ $S_4$ (as 4T5) trivial $5.12070505582$
4.4.13676.1 $x^{4} - x^{3} - 6 x^{2} + 7 x + 1$ $2^{2}\cdot 13\cdot 263$ $S_4$ (as 4T5) trivial $9.80390647748$
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