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Results (4 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
82.1-a1 82.1-a \(\Q(\sqrt{2}) \) \( 2 \cdot 41 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $0.651371891$ 1.151473703 \( -\frac{5814814559175147}{231712402} a - \frac{4128881686911505}{115856201} \) \( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -15 a - 79\) , \( -47 a - 251\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-15a-79\right){x}-47a-251$
82.1-a2 82.1-a \(\Q(\sqrt{2}) \) \( 2 \cdot 41 \) 0 $\Z/5\Z$ $\mathrm{SU}(2)$ $1$ $16.28429728$ 1.151473703 \( \frac{89373}{328} a + \frac{148955}{164} \) \( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 1\) , \( -a - 1\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+{x}-a-1$
82.1-b1 82.1-b \(\Q(\sqrt{2}) \) \( 2 \cdot 41 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $1.872522881$ 0.662036813 \( -\frac{25086047287643}{141150208} a - \frac{17749977458361}{70575104} \) \( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -7 a - 33\) , \( -97 a + 10\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-7a-33\right){x}-97a+10$
82.1-b2 82.1-b \(\Q(\sqrt{2}) \) \( 2 \cdot 41 \) 0 $\Z/3\Z$ $\mathrm{SU}(2)$ $1$ $16.85270592$ 0.662036813 \( \frac{64288010347}{656} a - \frac{45458394669}{328} \) \( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 13 a - 18\) , \( -30 a + 42\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(13a-18\right){x}-30a+42$
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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.