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Base field Conductor norm Rank* Torsion order CM discriminant
Base field degree/signature Bad \(p\) $\Q$-curves Torsion structure CM
Analytic order of Ш* Cyclic isogeny degree Isogeny class size Reduction Galois image
j-invariant Regulator* Curves per isogeny class Isogeny class degree Nonmax$\ \ell$
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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
15.1-a1 15.1-a 5.5.65657.1 \( 3 \cdot 5 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $396.8489402$ 1.54876208 \( \frac{43762711342084459813388}{6328125} a^{4} + \frac{19657683179991310734076}{2109375} a^{3} - \frac{80370595538065362656194}{6328125} a^{2} - \frac{11238856761605347645073}{703125} a - \frac{18641764309297167053009}{6328125} \) \( \bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 6\) , \( -2 a^{4} + 3 a^{3} + 8 a^{2} - 7 a - 5\) , \( -a^{4} + a^{3} + 5 a^{2} - 2 a - 4\) , \( -46 a^{4} + 109 a^{3} + 43 a^{2} - 117 a - 48\) , \( 463 a^{4} - 1139 a^{3} - 594 a^{2} + 1290 a + 662\bigr] \) ${y}^2+\left(-2a^{4}+3a^{3}+9a^{2}-9a-6\right){x}{y}+\left(-a^{4}+a^{3}+5a^{2}-2a-4\right){y}={x}^{3}+\left(-2a^{4}+3a^{3}+8a^{2}-7a-5\right){x}^{2}+\left(-46a^{4}+109a^{3}+43a^{2}-117a-48\right){x}+463a^{4}-1139a^{3}-594a^{2}+1290a+662$
15.1-a2 15.1-a 5.5.65657.1 \( 3 \cdot 5 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1587.395761$ 1.54876208 \( \frac{810387038438023516}{54931640625} a^{4} + \frac{363395431073753132}{18310546875} a^{3} - \frac{1491237718490098883}{54931640625} a^{2} - \frac{207634903981355686}{6103515625} a - \frac{338342690307372463}{54931640625} \) \( \bigl[a^{2} - 2\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{2} - 2\) , \( 10 a^{4} - 10 a^{3} - 40 a^{2} + 16 a + 5\) , \( -25 a^{4} + 18 a^{3} + 100 a^{2} - 13 a - 9\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){x}^{2}+\left(10a^{4}-10a^{3}-40a^{2}+16a+5\right){x}-25a^{4}+18a^{3}+100a^{2}-13a-9$
15.1-a3 15.1-a 5.5.65657.1 \( 3 \cdot 5 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1587.395761$ 1.54876208 \( \frac{1404373231}{234375} a^{4} - \frac{1413777188}{78125} a^{3} - \frac{830144678}{234375} a^{2} + \frac{1478965772}{78125} a + \frac{1473768167}{234375} \) \( \bigl[a^{2} - 2\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{2} - 2\) , \( a\) , \( -a^{4} + a^{3} + 4 a^{2} - a - 1\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){x}^{2}+a{x}-a^{4}+a^{3}+4a^{2}-a-1$
15.1-a4 15.1-a 5.5.65657.1 \( 3 \cdot 5 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $198.4244701$ 1.54876208 \( \frac{359045212142585076290656747468}{111758708953857421875} a^{4} - \frac{147682742439513102209920029764}{37252902984619140625} a^{3} - \frac{1691569060032060442379194043234}{111758708953857421875} a^{2} + \frac{371284744650378795256836683741}{37252902984619140625} a + \frac{1534625546447161055611719609551}{111758708953857421875} \) \( \bigl[a^{2} - 2\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 2\) , \( a^{2} - 2\) , \( 20 a^{4} - 20 a^{3} - 75 a^{2} + 26 a\) , \( 30 a^{4} - 29 a^{3} - 120 a^{2} + 39 a + 19\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{4}-a^{3}-4a^{2}+2a+2\right){x}^{2}+\left(20a^{4}-20a^{3}-75a^{2}+26a\right){x}+30a^{4}-29a^{3}-120a^{2}+39a+19$
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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.