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Results (8 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
68.1-a1 68.1-a \(\Q(\sqrt{85}) \) \( 2^{2} \cdot 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $20.21098874$ 6.576568561 \( \frac{3048625}{1088} \) \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 31 a - 176\) , \( -80 a + 399\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(31a-176\right){x}-80a+399$
68.1-a2 68.1-a \(\Q(\sqrt{85}) \) \( 2^{2} \cdot 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $2.245665415$ 6.576568561 \( \frac{159661140625}{48275138} \) \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 1241 a - 6446\) , \( 38090 a - 194961\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1241a-6446\right){x}+38090a-194961$
68.1-a3 68.1-a \(\Q(\sqrt{85}) \) \( 2^{2} \cdot 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $20.21098874$ 6.576568561 \( \frac{8805624625}{2312} \) \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 471 a - 2456\) , \( -11288 a + 57711\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(471a-2456\right){x}-11288a+57711$
68.1-a4 68.1-a \(\Q(\sqrt{85}) \) \( 2^{2} \cdot 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $2.245665415$ 6.576568561 \( \frac{120920208625}{19652} \) \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 1131 a - 5876\) , \( 47164 a - 241377\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1131a-5876\right){x}+47164a-241377$
68.1-b1 68.1-b \(\Q(\sqrt{85}) \) \( 2^{2} \cdot 17 \) $1$ $\Z/6\Z$ $\mathrm{SU}(2)$ $0.560163422$ $20.21098874$ 0.818656255 \( \frac{3048625}{1088} \) \( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \) ${y}^2+{x}{y}={x}^{3}-3{x}+1$
68.1-b2 68.1-b \(\Q(\sqrt{85}) \) \( 2^{2} \cdot 17 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $3.360980532$ $2.245665415$ 0.818656255 \( \frac{159661140625}{48275138} \) \( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \) ${y}^2+{x}{y}={x}^{3}-113{x}-329$
68.1-b3 68.1-b \(\Q(\sqrt{85}) \) \( 2^{2} \cdot 17 \) $1$ $\Z/6\Z$ $\mathrm{SU}(2)$ $1.120326844$ $20.21098874$ 0.818656255 \( \frac{8805624625}{2312} \) \( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \) ${y}^2+{x}{y}={x}^{3}-43{x}+105$
68.1-b4 68.1-b \(\Q(\sqrt{85}) \) \( 2^{2} \cdot 17 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1.680490266$ $2.245665415$ 0.818656255 \( \frac{120920208625}{19652} \) \( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \) ${y}^2+{x}{y}={x}^{3}-103{x}-411$
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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.