Elliptic curve isogeny class 648.1-f over number field \(\Q(\sqrt{7}) \)

• Base field: \(\Q(\sqrt{7}) \)
• Label: 2.2.28.1-648.1-f
• Conductor: 648.1
• Rank: \( 1 \)

Base field \(\Q(\sqrt{7}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 7 \); class number \(1\).

Elliptic curves in class 648.1-f over \(\Q(\sqrt{7}) \)

Isogeny class 648.1-f contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
648.1-f1	\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -1690 a + 4481\) , \( 451707 a - 1195095\bigr] \)
648.1-f2	\( \bigl[0\) , \( 0\) , \( 0\) , \( 6\) , \( 7\bigr] \)
648.1-f3	\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -465 a - 1230\) , \( 5609 a + 14840\bigr] \)
648.1-f4	\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2630 a - 6949\) , \( 112947 a - 298821\bigr] \)
648.1-f5	\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -6945 a - 18375\) , \( 496604 a + 1313891\bigr] \)
648.1-f6	\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -41505 a - 109815\) , \( -7584046 a - 20065501\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 8 & 4 \\ 8 & 1 & 2 & 4 & 4 & 8 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 2 & 4 & 2 & 1 & 4 & 2 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 8 & 4 & 2 & 8 & 1 \end{array}\right)\)

Isogeny graph