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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 (5 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
32.1-a1 32.1-a 5.5.70601.1 \( 2^{5} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.801650473$ $176.9150737$ 2.66879041 \( -\frac{2485641}{2} a^{4} + 1159797 a^{3} + 4981591 a^{2} - 2602560 a - 1253263 \) \( \bigl[a^{4} - a^{3} - 4 a^{2} + 2 a + 1\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 4 a\) , \( 2 a^{4} - a^{3} - 11 a^{2} + a + 6\) , \( -3 a^{3} + 5 a^{2} + 9 a - 9\) , \( -3 a^{4} + a^{3} + 16 a^{2} + 4 a - 13\bigr] \) ${y}^2+\left(a^{4}-a^{3}-4a^{2}+2a+1\right){x}{y}+\left(2a^{4}-a^{3}-11a^{2}+a+6\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+4a\right){x}^{2}+\left(-3a^{3}+5a^{2}+9a-9\right){x}-3a^{4}+a^{3}+16a^{2}+4a-13$
32.1-b1 32.1-b 5.5.70601.1 \( 2^{5} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.011385182$ $12116.38451$ 2.59583819 \( \frac{141897}{2} a^{4} - 47459 a^{3} - 370176 a^{2} + \frac{38961}{2} a + \frac{435239}{2} \) \( \bigl[2 a^{4} - a^{3} - 11 a^{2} + a + 5\) , \( -a^{4} + a^{3} + 5 a^{2} - 3 a - 3\) , \( 3 a^{4} - 2 a^{3} - 15 a^{2} + a + 5\) , \( 9 a^{4} - 4 a^{3} - 39 a^{2} - 20 a + 3\) , \( -31 a^{4} + a^{3} + 170 a^{2} + 75 a - 45\bigr] \) ${y}^2+\left(2a^{4}-a^{3}-11a^{2}+a+5\right){x}{y}+\left(3a^{4}-2a^{3}-15a^{2}+a+5\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-3a-3\right){x}^{2}+\left(9a^{4}-4a^{3}-39a^{2}-20a+3\right){x}-31a^{4}+a^{3}+170a^{2}+75a-45$
32.1-c1 32.1-c 5.5.70601.1 \( 2^{5} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.023197209$ $5637.474343$ 2.46084863 \( -\frac{32857413}{2} a^{4} - \frac{99317097}{4} a^{3} + \frac{79437623}{4} a^{2} + 17064543 a - \frac{52421865}{8} \) \( \bigl[a^{4} - a^{3} - 5 a^{2} + 2 a + 2\) , \( -a^{4} + 5 a^{2} + 3 a - 2\) , \( a\) , \( -7 a^{4} + a^{3} + 37 a^{2} + 17 a - 11\) , \( 2 a^{4} - 11 a^{2} - 6 a + 3\bigr] \) ${y}^2+\left(a^{4}-a^{3}-5a^{2}+2a+2\right){x}{y}+a{y}={x}^{3}+\left(-a^{4}+5a^{2}+3a-2\right){x}^{2}+\left(-7a^{4}+a^{3}+37a^{2}+17a-11\right){x}+2a^{4}-11a^{2}-6a+3$
32.1-c2 32.1-c 5.5.70601.1 \( 2^{5} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.007732403$ $16912.42303$ 2.46084863 \( 410269356 a^{4} - 278026353 a^{3} - 2140972273 a^{2} + 130455752 a + \frac{2545730995}{2} \) \( \bigl[2 a^{4} - a^{3} - 11 a^{2} + 5\) , \( 2 a^{4} - a^{3} - 10 a^{2} - 2 a + 4\) , \( a^{4} - a^{3} - 4 a^{2} + a\) , \( 9 a^{4} - 5 a^{3} - 44 a^{2} - 8 a + 19\) , \( -11 a^{4} + 58 a^{2} + 32 a - 9\bigr] \) ${y}^2+\left(2a^{4}-a^{3}-11a^{2}+5\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+a\right){y}={x}^{3}+\left(2a^{4}-a^{3}-10a^{2}-2a+4\right){x}^{2}+\left(9a^{4}-5a^{3}-44a^{2}-8a+19\right){x}-11a^{4}+58a^{2}+32a-9$
32.1-d1 32.1-d 5.5.70601.1 \( 2^{5} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.015698445$ $8567.434520$ 2.53088274 \( \frac{454837}{2} a^{4} - 635149 a^{3} + \frac{3677}{2} a^{2} + \frac{902141}{2} a - 126645 \) \( \bigl[a\) , \( -a^{4} + 5 a^{2} + 3 a - 2\) , \( a^{4} - 6 a^{2} - a + 4\) , \( -2 a^{4} + a^{3} + 9 a^{2} + 2 a - 2\) , \( -336 a^{4} + 64 a^{3} + 1731 a^{2} + 730 a - 416\bigr] \) ${y}^2+a{x}{y}+\left(a^{4}-6a^{2}-a+4\right){y}={x}^{3}+\left(-a^{4}+5a^{2}+3a-2\right){x}^{2}+\left(-2a^{4}+a^{3}+9a^{2}+2a-2\right){x}-336a^{4}+64a^{3}+1731a^{2}+730a-416$
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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.