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Results (30 matches)

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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
23409.8-a1 23409.8-a \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\mathsf{trivial}$ $0.069480281$ $4.574307975$ 1.797885198 \( -\frac{110592}{17} \) \( \bigl[0\) , \( 0\) , \( 1\) , \( -3\) , \( 2\bigr] \) ${y}^2+{y}={x}^{3}-3{x}+2$
23409.8-b1 23409.8-b \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $0.463200697$ $0.930162799$ 2.437267295 \( -\frac{4375968098}{1193859} a - \frac{83257760131}{1193859} \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 82 a - 26\) , \( -306 a - 318\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(82a-26\right){x}-306a-318$
23409.8-b2 23409.8-b \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $0.926401395$ $0.465081399$ 2.437267295 \( \frac{454229228687227}{1425299311881} a + \frac{679301259838487}{1425299311881} \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 67 a - 131\) , \( -105 a - 1206\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(67a-131\right){x}-105a-1206$
23409.8-c1 23409.8-c \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $1.255563294$ $0.147713997$ 3.147433083 \( -\frac{372082589114986904}{2610969633} a - \frac{181318779827784209}{2610969633} \) \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 9785 a + 827\) , \( 287614 a + 472652\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9785a+827\right){x}+287614a+472652$
23409.8-c2 23409.8-c \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.627781647$ $0.295427994$ 3.147433083 \( \frac{147770521717808}{17596287801} a - \frac{136210703475223}{17596287801} \) \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 605 a + 62\) , \( 4870 a + 6920\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(605a+62\right){x}+4870a+6920$
23409.8-c3 23409.8-c \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $1.255563294$ $0.147713997$ 3.147433083 \( \frac{1695948356939509768}{15730800405203547} a - \frac{7330062339265374821}{15730800405203547} \) \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 65 a - 1423\) , \( 118 a + 43424\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(65a-1423\right){x}+118a+43424$
23409.8-c4 23409.8-c \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/4\Z$ $0.313890823$ $0.590855988$ 3.147433083 \( -\frac{3780522976}{2255067} a + \frac{417620747}{2255067} \) \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 65 a + 107\) , \( 442 a - 514\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(65a+107\right){x}+442a-514$
23409.8-d1 23409.8-d \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $0.463200697$ $0.930162799$ 2.437267295 \( \frac{4375968098}{1193859} a - \frac{83257760131}{1193859} \) \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -82 a - 25\) , \( 281 a - 154\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-82a-25\right){x}+281a-154$
23409.8-d2 23409.8-d \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $0.926401395$ $0.465081399$ 2.437267295 \( -\frac{454229228687227}{1425299311881} a + \frac{679301259838487}{1425299311881} \) \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -67 a - 130\) , \( -25 a - 1072\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-67a-130\right){x}-25a-1072$
23409.8-e1 23409.8-e \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $1.255563294$ $0.147713997$ 3.147433083 \( \frac{372082589114986904}{2610969633} a - \frac{181318779827784209}{2610969633} \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -9785 a + 826\) , \( -286787 a + 492222\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-9785a+826\right){x}-286787a+492222$
23409.8-e2 23409.8-e \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.627781647$ $0.295427994$ 3.147433083 \( -\frac{147770521717808}{17596287801} a - \frac{136210703475223}{17596287801} \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -605 a + 61\) , \( -4808 a + 8130\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-605a+61\right){x}-4808a+8130$
23409.8-e3 23409.8-e \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $1.255563294$ $0.147713997$ 3.147433083 \( -\frac{1695948356939509768}{15730800405203547} a - \frac{7330062339265374821}{15730800405203547} \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -65 a - 1424\) , \( -1541 a + 43554\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-65a-1424\right){x}-1541a+43554$
23409.8-e4 23409.8-e \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/4\Z$ $0.313890823$ $0.590855988$ 3.147433083 \( \frac{3780522976}{2255067} a + \frac{417620747}{2255067} \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -65 a + 106\) , \( -335 a - 384\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-65a+106\right){x}-335a-384$
23409.8-f1 23409.8-f \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\Z/2\Z$ $1$ $0.507090095$ 0.717133690 \( -\frac{1875101184}{24137569} a - \frac{11541230784}{24137569} \) \( \bigl[a\) , \( -1\) , \( 1\) , \( 67 a + 75\) , \( 189 a - 1077\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(67a+75\right){x}+189a-1077$
23409.8-f2 23409.8-f \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\Z/6\Z$ $1$ $1.521270287$ 0.717133690 \( \frac{1875101184}{24137569} a - \frac{11541230784}{24137569} \) \( \bigl[a\) , \( -1\) , \( 1\) , \( -8 a + 9\) , \( 12 a + 34\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-8a+9\right){x}+12a+34$
23409.8-f3 23409.8-f \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\Z/6\Z$ $1$ $1.521270287$ 0.717133690 \( -\frac{937944576}{4913} a + \frac{5404256064}{4913} \) \( \bigl[a\) , \( -1\) , \( 1\) , \( -38 a + 45\) , \( -21 a - 236\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-38a+45\right){x}-21a-236$
23409.8-f4 23409.8-f \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\Z/2\Z$ $1$ $0.507090095$ 0.717133690 \( \frac{937944576}{4913} a + \frac{5404256064}{4913} \) \( \bigl[a\) , \( -1\) , \( 1\) , \( 337 a + 399\) , \( -1242 a + 5565\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(337a+399\right){x}-1242a+5565$
23409.8-g1 23409.8-g \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\Z/6\Z$ $1$ $1.521270287$ 0.717133690 \( -\frac{1875101184}{24137569} a - \frac{11541230784}{24137569} \) \( \bigl[a\) , \( -1\) , \( 1\) , \( 7 a + 9\) , \( -12 a + 34\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(7a+9\right){x}-12a+34$
23409.8-g2 23409.8-g \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\Z/2\Z$ $1$ $0.507090095$ 0.717133690 \( \frac{1875101184}{24137569} a - \frac{11541230784}{24137569} \) \( \bigl[a\) , \( -1\) , \( 1\) , \( -68 a + 75\) , \( -189 a - 1077\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-68a+75\right){x}-189a-1077$
23409.8-g3 23409.8-g \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\Z/2\Z$ $1$ $0.507090095$ 0.717133690 \( -\frac{937944576}{4913} a + \frac{5404256064}{4913} \) \( \bigl[a\) , \( -1\) , \( 1\) , \( -338 a + 399\) , \( 1242 a + 5565\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-338a+399\right){x}+1242a+5565$
23409.8-g4 23409.8-g \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\Z/6\Z$ $1$ $1.521270287$ 0.717133690 \( \frac{937944576}{4913} a + \frac{5404256064}{4913} \) \( \bigl[a\) , \( -1\) , \( 1\) , \( 37 a + 45\) , \( 21 a - 236\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(37a+45\right){x}+21a-236$
23409.8-h1 23409.8-h \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/3\Z$ $0.338669215$ $0.559805910$ 4.289910009 \( -\frac{23100424192}{14739} \) \( \bigl[0\) , \( 0\) , \( 1\) , \( -534\) , \( 4752\bigr] \) ${y}^2+{y}={x}^{3}-534{x}+4752$
23409.8-h2 23409.8-h \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\mathsf{trivial}$ $0.112889738$ $1.679417732$ 4.289910009 \( \frac{32768}{459} \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 6\) , \( 27\bigr] \) ${y}^2+{y}={x}^{3}+6{x}+27$
23409.8-i1 23409.8-i \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $1.247582370$ $0.707979566$ 4.996489067 \( -\frac{35937}{83521} \) \( \bigl[1\) , \( -1\) , \( 0\) , \( -6\) , \( 377\bigr] \) ${y}^2+{x}{y}={x}^{3}-{x}^{2}-6{x}+377$
23409.8-i2 23409.8-i \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $1.247582370$ $2.831918265$ 4.996489067 \( \frac{35937}{17} \) \( \bigl[1\) , \( -1\) , \( 0\) , \( -6\) , \( -1\bigr] \) ${y}^2+{x}{y}={x}^{3}-{x}^{2}-6{x}-1$
23409.8-i3 23409.8-i \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.623791185$ $1.415959132$ 4.996489067 \( \frac{20346417}{289} \) \( \bigl[1\) , \( -1\) , \( 0\) , \( -51\) , \( 152\bigr] \) ${y}^2+{x}{y}={x}^{3}-{x}^{2}-51{x}+152$
23409.8-i4 23409.8-i \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\Z/2\Z$ $1.247582370$ $0.707979566$ 4.996489067 \( \frac{82483294977}{17} \) \( \bigl[1\) , \( -1\) , \( 0\) , \( -816\) , \( 9179\bigr] \) ${y}^2+{x}{y}={x}^{3}-{x}^{2}-816{x}+9179$
23409.8-j1 23409.8-j \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\mathsf{trivial}$ $1$ $0.721454620$ 4.081163634 \( \frac{14811123712}{96702579} a + \frac{26737709056}{96702579} \) \( \bigl[0\) , \( 0\) , \( a + 1\) , \( 27 a - 39\) , \( -a - 317\bigr] \) ${y}^2+\left(a+1\right){y}={x}^{3}+\left(27a-39\right){x}-a-317$
23409.8-k1 23409.8-k \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $0$ $\mathsf{trivial}$ $1$ $0.721454620$ 4.081163634 \( -\frac{14811123712}{96702579} a + \frac{26737709056}{96702579} \) \( \bigl[0\) , \( 0\) , \( a + 1\) , \( -27 a - 39\) , \( -317\bigr] \) ${y}^2+\left(a+1\right){y}={x}^{3}+\left(-27a-39\right){x}-317$
23409.8-l1 23409.8-l \(\Q(\sqrt{-2}) \) \( 3^{4} \cdot 17^{2} \) $1$ $\mathsf{trivial}$ $1.159378227$ $1.524769325$ 10.00009845 \( -\frac{110592}{17} \) \( \bigl[0\) , \( 0\) , \( 1\) , \( -27\) , \( -61\bigr] \) ${y}^2+{y}={x}^{3}-27{x}-61$
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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.