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The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 4091 over totally real quartic fields with discriminant 19821

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Base field Conductor norm Rank* Torsion order CM discriminant
Base field degree/signature Bad \(p\) $\Q$-curves Torsion structure CM
Analytic order of Ш* Cyclic isogeny degree Isogeny class size Reduction Galois image
j-invariant Regulator* Curves per isogeny class Isogeny class degree Nonmax$\ \ell$
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Results (12 matches)

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Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
31.1-a1 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $226.2402485$ 0.843147624 \( \frac{13262138608154212980818020352}{923521} a^{3} - \frac{16034680544860826249617094257}{923521} a^{2} - \frac{49696393399932989894285277456}{923521} a + \frac{63437929696698697519404008684}{923521} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -427 a^{3} - 156 a^{2} + 1210 a - 328\) , \( 5301 a^{3} + 2573 a^{2} - 14446 a + 2048\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-427a^{3}-156a^{2}+1210a-328\right){x}+5301a^{3}+2573a^{2}-14446a+2048$
31.1-a2 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $14.14001553$ 0.843147624 \( -\frac{69643673441152672038181763072}{727423121747185263828481} a^{3} + \frac{84890779633537182060967675121}{727423121747185263828481} a^{2} + \frac{261357927431140745378290491664}{727423121747185263828481} a - \frac{334879646814468933111035546908}{727423121747185263828481} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( 113 a^{3} + 114 a^{2} - 280 a - 98\) , \( -a^{3} - 75 a^{2} + 44 a + 78\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(113a^{3}+114a^{2}-280a-98\right){x}-a^{3}-75a^{2}+44a+78$
31.1-a3 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $226.2402485$ 0.843147624 \( \frac{43561848213205037510145}{852891037441} a^{3} - \frac{52668771245192606365444}{852891037441} a^{2} - \frac{163236628849912597561348}{852891037441} a + \frac{208373205080817133396719}{852891037441} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -37 a^{3} - 21 a^{2} + 105 a - 13\) , \( 2 a^{3} - 25 a^{2} - 17 a + 63\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-37a^{3}-21a^{2}+105a-13\right){x}+2a^{3}-25a^{2}-17a+63$
31.1-a4 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $226.2402485$ 0.843147624 \( -\frac{119292}{961} a^{3} - \frac{1134908}{961} a^{2} + \frac{2242557}{961} a + \frac{309574}{961} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -2 a^{3} - a^{2} + 5 a + 2\) , \( -2 a^{3} - 2 a^{2} + 5 a + 1\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-2a^{3}-a^{2}+5a+2\right){x}-2a^{3}-2a^{2}+5a+1$
31.1-a5 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $226.2402485$ 0.843147624 \( -\frac{2933497938285}{923521} a^{3} + \frac{9133676668039}{923521} a^{2} - \frac{6067822613957}{923521} a - \frac{1204128338708}{923521} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -22 a^{3} - 21 a^{2} + 60 a + 12\) , \( -77 a^{3} - 67 a^{2} + 195 a + 43\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-22a^{3}-21a^{2}+60a+12\right){x}-77a^{3}-67a^{2}+195a+43$
31.1-a6 31.1-a \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $14.14001553$ 0.843147624 \( -\frac{236284316384699073}{961} a^{3} + \frac{698632890478410244}{961} a^{2} - \frac{421506652954098940}{961} a - \frac{120811375486877791}{961} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 1\) , \( -327 a^{3} - 341 a^{2} + 895 a + 197\) , \( -4716 a^{3} - 4369 a^{2} + 12267 a + 2711\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-327a^{3}-341a^{2}+895a+197\right){x}-4716a^{3}-4369a^{2}+12267a+2711$
31.1-b1 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $2.959428063$ 0.705864781 \( \frac{13262138608154212980818020352}{923521} a^{3} - \frac{16034680544860826249617094257}{923521} a^{2} - \frac{49696393399932989894285277456}{923521} a + \frac{63437929696698697519404008684}{923521} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -80 a^{3} + 335 a^{2} + 315 a - 1284\) , \( -2177 a^{3} + 4753 a^{2} + 8378 a - 18117\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-80a^{3}+335a^{2}+315a-1284\right){x}-2177a^{3}+4753a^{2}+8378a-18117$
31.1-b2 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $47.35084901$ 0.705864781 \( \frac{43561848213205037510145}{852891037441} a^{3} - \frac{52668771245192606365444}{852891037441} a^{2} - \frac{163236628849912597561348}{852891037441} a + \frac{208373205080817133396719}{852891037441} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -5 a^{3} + 20 a^{2} + 20 a - 79\) , \( -42 a^{3} + 85 a^{2} + 161 a - 317\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-5a^{3}+20a^{2}+20a-79\right){x}-42a^{3}+85a^{2}+161a-317$
31.1-b3 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $2.959428063$ 0.705864781 \( -\frac{69643673441152672038181763072}{727423121747185263828481} a^{3} + \frac{84890779633537182060967675121}{727423121747185263828481} a^{2} + \frac{261357927431140745378290491664}{727423121747185263828481} a - \frac{334879646814468933111035546908}{727423121747185263828481} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -10 a^{3} + 25 a^{2} + 45 a - 74\) , \( 9 a^{3} + 101 a^{2} - 28 a - 357\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-10a^{3}+25a^{2}+45a-74\right){x}+9a^{3}+101a^{2}-28a-357$
31.1-b4 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/8\Z$ $\mathrm{SU}(2)$ $1$ $757.6135842$ 0.705864781 \( -\frac{236284316384699073}{961} a^{3} + \frac{698632890478410244}{961} a^{2} - \frac{421506652954098940}{961} a - \frac{120811375486877791}{961} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( 5 a^{3} - 20 a^{2} - 20 a - 9\) , \( -4 a^{3} + 99 a^{2} + 23 a - 5\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(5a^{3}-20a^{2}-20a-9\right){x}-4a^{3}+99a^{2}+23a-5$
31.1-b5 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/2\Z\oplus\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $757.6135842$ 0.705864781 \( -\frac{2933497938285}{923521} a^{3} + \frac{9133676668039}{923521} a^{2} - \frac{6067822613957}{923521} a - \frac{1204128338708}{923521} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -4\) , \( -a^{3} + 4 a^{2} + 4 a - 7\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}-4{x}-a^{3}+4a^{2}+4a-7$
31.1-b6 31.1-b \(\Q(\zeta_{15})^+\) \( 31 \) 0 $\Z/8\Z$ $\mathrm{SU}(2)$ $1$ $757.6135842$ 0.705864781 \( -\frac{119292}{961} a^{3} - \frac{1134908}{961} a^{2} + \frac{2242557}{961} a + \frac{309574}{961} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+{x}$
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  *The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.