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Results (6 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
1681.5-a1 1681.5-a \(\Q(\zeta_{7})^+\) \( 41^{2} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $5.387838465$ $0.826702528$ 1.908917005 \( \frac{174656123697499408263335}{13422659310152401} a^{2} + \frac{182915726357803972950650}{13422659310152401} a - \frac{115913951592431810832436}{13422659310152401} \) \( \bigl[a^{2} - 1\) , \( a^{2} - a - 1\) , \( a^{2} - 1\) , \( -1272 a^{2} - 1375 a - 299\) , \( 45529 a^{2} + 47686 a - 16832\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-1272a^{2}-1375a-299\right){x}+45529a^{2}+47686a-16832$
1681.5-a2 1681.5-a \(\Q(\zeta_{7})^+\) \( 41^{2} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $2.693919232$ $1.653405056$ 1.908917005 \( \frac{653981503916958524755}{115856201} a^{2} - \frac{1469483101129546552831}{115856201} a + \frac{524452446825637320235}{115856201} \) \( \bigl[a^{2} - 1\) , \( a^{2} - a - 1\) , \( a^{2} - 1\) , \( -267 a^{2} + 195 a + 96\) , \( -707 a^{2} + 3324 a - 255\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-267a^{2}+195a+96\right){x}-707a^{2}+3324a-255$
1681.5-a3 1681.5-a \(\Q(\zeta_{7})^+\) \( 41^{2} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.538783846$ $8.267025280$ 1.908917005 \( \frac{1703161}{41} a^{2} - \frac{968480}{41} a - \frac{3784992}{41} \) \( \bigl[a\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( -6 a^{2} + 15 a - 7\) , \( -3 a^{2} + 2 a - 2\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-6a^{2}+15a-7\right){x}-3a^{2}+2a-2$
1681.5-a4 1681.5-a \(\Q(\zeta_{7})^+\) \( 41^{2} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1.077567693$ $4.133512640$ 1.908917005 \( -\frac{6655766653200}{1681} a^{2} + \frac{3693705667625}{1681} a + \frac{14955417009784}{1681} \) \( \bigl[a\) , \( -a^{2} - a + 1\) , \( a^{2} + a - 2\) , \( 29 a^{2} - 75 a - 2\) , \( 29 a^{2} - 64 a - 45\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(29a^{2}-75a-2\right){x}+29a^{2}-64a-45$
1681.5-b1 1681.5-b \(\Q(\zeta_{7})^+\) \( 41^{2} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $9.867867490$ 1.409695355 \( 6041 a^{2} - 2135 a - 11897 \) \( \bigl[a\) , \( 1\) , \( a^{2} + a - 1\) , \( 4 a^{2} - 9 a - 3\) , \( 16 a^{2} - 35 a + 3\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+{x}^{2}+\left(4a^{2}-9a-3\right){x}+16a^{2}-35a+3$
1681.5-c1 1681.5-c \(\Q(\zeta_{7})^+\) \( 41^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.017805060$ $242.1147169$ 1.847514482 \( 6041 a^{2} - 2135 a - 11897 \) \( \bigl[1\) , \( -a\) , \( a^{2} - 1\) , \( a^{2} - a - 2\) , \( -a^{2} + 2\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-1\right){y}={x}^{3}-a{x}^{2}+\left(a^{2}-a-2\right){x}-a^{2}+2$
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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.