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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
29.3-a1 29.3-a \(\Q(\sqrt{5}, \sqrt{7})\) \( 29 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.061084943$ $501.7034145$ 2.626844961 \( \frac{26708755}{560947} a^{3} - \frac{78978730}{560947} a^{2} - \frac{312565105}{560947} a + \frac{20685109}{19343} \) \( \bigl[-\frac{3}{23} a^{3} + \frac{16}{23} a^{2} + \frac{30}{23} a - \frac{102}{23}\) , \( -a + 1\) , \( -\frac{3}{23} a^{3} + \frac{16}{23} a^{2} + \frac{30}{23} a - \frac{125}{23}\) , \( \frac{118}{23} a^{3} - \frac{361}{23} a^{2} - \frac{1433}{23} a + \frac{3414}{23}\) , \( \frac{594}{23} a^{3} - \frac{1811}{23} a^{2} - \frac{6998}{23} a + \frac{16677}{23}\bigr] \) ${y}^2+\left(-\frac{3}{23}a^{3}+\frac{16}{23}a^{2}+\frac{30}{23}a-\frac{102}{23}\right){x}{y}+\left(-\frac{3}{23}a^{3}+\frac{16}{23}a^{2}+\frac{30}{23}a-\frac{125}{23}\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(\frac{118}{23}a^{3}-\frac{361}{23}a^{2}-\frac{1433}{23}a+\frac{3414}{23}\right){x}+\frac{594}{23}a^{3}-\frac{1811}{23}a^{2}-\frac{6998}{23}a+\frac{16677}{23}$
29.3-a2 29.3-a \(\Q(\sqrt{5}, \sqrt{7})\) \( 29 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.183254829$ $501.7034145$ 2.626844961 \( -\frac{2260418171}{667} a^{3} - \frac{5116700339}{667} a^{2} + \frac{12088543944}{667} a + \frac{530057237}{23} \) \( \bigl[-\frac{4}{23} a^{3} + \frac{6}{23} a^{2} + \frac{63}{23} a - \frac{44}{23}\) , \( \frac{1}{23} a^{3} - \frac{13}{23} a^{2} - \frac{10}{23} a + \frac{126}{23}\) , \( -\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a + \frac{1}{23}\) , \( -\frac{13}{23} a^{3} - \frac{61}{23} a^{2} + \frac{245}{23} a + \frac{961}{23}\) , \( \frac{162}{23} a^{3} - \frac{59}{23} a^{2} - \frac{2333}{23} a - \frac{1369}{23}\bigr] \) ${y}^2+\left(-\frac{4}{23}a^{3}+\frac{6}{23}a^{2}+\frac{63}{23}a-\frac{44}{23}\right){x}{y}+\left(-\frac{2}{23}a^{3}+\frac{3}{23}a^{2}+\frac{43}{23}a+\frac{1}{23}\right){y}={x}^{3}+\left(\frac{1}{23}a^{3}-\frac{13}{23}a^{2}-\frac{10}{23}a+\frac{126}{23}\right){x}^{2}+\left(-\frac{13}{23}a^{3}-\frac{61}{23}a^{2}+\frac{245}{23}a+\frac{961}{23}\right){x}+\frac{162}{23}a^{3}-\frac{59}{23}a^{2}-\frac{2333}{23}a-\frac{1369}{23}$
29.3-b1 29.3-b \(\Q(\sqrt{5}, \sqrt{7})\) \( 29 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.175792773$ $217.9490740$ 3.284046195 \( \frac{26708755}{560947} a^{3} - \frac{78978730}{560947} a^{2} - \frac{312565105}{560947} a + \frac{20685109}{19343} \) \( \bigl[-\frac{1}{23} a^{3} + \frac{13}{23} a^{2} - \frac{13}{23} a - \frac{103}{23}\) , \( \frac{4}{23} a^{3} - \frac{6}{23} a^{2} - \frac{63}{23} a + \frac{21}{23}\) , \( 1\) , \( \frac{122}{23} a^{3} - \frac{367}{23} a^{2} - \frac{1473}{23} a + \frac{3527}{23}\) , \( -\frac{40}{23} a^{3} + \frac{129}{23} a^{2} + \frac{423}{23} a - \frac{1061}{23}\bigr] \) ${y}^2+\left(-\frac{1}{23}a^{3}+\frac{13}{23}a^{2}-\frac{13}{23}a-\frac{103}{23}\right){x}{y}+{y}={x}^{3}+\left(\frac{4}{23}a^{3}-\frac{6}{23}a^{2}-\frac{63}{23}a+\frac{21}{23}\right){x}^{2}+\left(\frac{122}{23}a^{3}-\frac{367}{23}a^{2}-\frac{1473}{23}a+\frac{3527}{23}\right){x}-\frac{40}{23}a^{3}+\frac{129}{23}a^{2}+\frac{423}{23}a-\frac{1061}{23}$
29.3-b2 29.3-b \(\Q(\sqrt{5}, \sqrt{7})\) \( 29 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.527378321$ $217.9490740$ 3.284046195 \( -\frac{2260418171}{667} a^{3} - \frac{5116700339}{667} a^{2} + \frac{12088543944}{667} a + \frac{530057237}{23} \) \( \bigl[-\frac{1}{23} a^{3} + \frac{13}{23} a^{2} + \frac{10}{23} a - \frac{80}{23}\) , \( \frac{1}{23} a^{3} + \frac{10}{23} a^{2} - \frac{10}{23} a - \frac{58}{23}\) , \( -\frac{3}{23} a^{3} + \frac{16}{23} a^{2} + \frac{30}{23} a - \frac{102}{23}\) , \( -\frac{21}{23} a^{3} + \frac{43}{23} a^{2} + \frac{417}{23} a + \frac{183}{23}\) , \( -\frac{166}{23} a^{3} + \frac{42}{23} a^{2} + \frac{2626}{23} a + \frac{2337}{23}\bigr] \) ${y}^2+\left(-\frac{1}{23}a^{3}+\frac{13}{23}a^{2}+\frac{10}{23}a-\frac{80}{23}\right){x}{y}+\left(-\frac{3}{23}a^{3}+\frac{16}{23}a^{2}+\frac{30}{23}a-\frac{102}{23}\right){y}={x}^{3}+\left(\frac{1}{23}a^{3}+\frac{10}{23}a^{2}-\frac{10}{23}a-\frac{58}{23}\right){x}^{2}+\left(-\frac{21}{23}a^{3}+\frac{43}{23}a^{2}+\frac{417}{23}a+\frac{183}{23}\right){x}-\frac{166}{23}a^{3}+\frac{42}{23}a^{2}+\frac{2626}{23}a+\frac{2337}{23}$
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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.