Properties

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

Related objects

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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