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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
51.1-a1 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1.476452004$ 0.656200891 \( -\frac{73067274470120883503414977}{188345450907} a^{2} + \frac{25375998117751504521013489}{188345450907} a + \frac{70129610583765315423118325}{62781816969} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( 417 a^{2} - 133 a - 1203\) , \( 4217 a^{2} - 237 a - 14449\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(417a^{2}-133a-1203\right){x}+4217a^{2}-237a-14449$
51.1-a2 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $47.24646415$ 0.656200891 \( \frac{9123267589576143071594}{153} a^{2} + \frac{17146134462759887246327}{153} a + \frac{4854389290588414014785}{153} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( 27 a^{2} - 58 a - 43\) , \( 50 a^{2} - 62 a + 9\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(27a^{2}-58a-43\right){x}+50a^{2}-62a+9$
51.1-a3 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $11.81161603$ 0.656200891 \( -\frac{736214157382250}{60886809} a^{2} + \frac{257154008212105}{60886809} a + \frac{2126010334493071}{60886809} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -33 a^{2} + 92 a - 33\) , \( -292 a^{2} + 546 a + 23\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-33a^{2}+92a-33\right){x}-292a^{2}+546a+23$
51.1-a4 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) 0 $\Z/2\Z\oplus\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $94.49292831$ 0.656200891 \( \frac{21862041663547}{7803} a^{2} + \frac{41087226515522}{7803} a + \frac{3877536454165}{2601} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -3 a^{2} + 17 a + 2\) , \( 11 a^{2} - 8 a - 6\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-3a^{2}+17a+2\right){x}+11a^{2}-8a-6$
51.1-a5 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) 0 $\Z/8\Z$ $\mathrm{SU}(2)$ $1$ $188.9858566$ 0.656200891 \( \frac{57227017}{153} a^{2} - \frac{97672967}{153} a - \frac{40772771}{153} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -3 a^{2} + 17 a + 7\) , \( 17 a^{2} - 13 a - 7\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-3a^{2}+17a+7\right){x}+17a^{2}-13a-7$
51.1-a6 51.1-a \(\Q(\zeta_{9})^+\) \( 3 \cdot 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1.476452004$ 0.656200891 \( \frac{51150856611467521}{51195483} a^{2} - \frac{78367545688483633}{51195483} a - \frac{3709618270320887}{5688387} \) \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -963 a^{2} + 1517 a + 577\) , \( -24013 a^{2} + 36905 a + 15491\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-963a^{2}+1517a+577\right){x}-24013a^{2}+36905a+15491$
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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.