Show commands:
SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 288990.ea
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
288990.ea1 | 288990ea4 | \([1, -1, 1, -704574383, -4596367959673]\) | \(10993009831928446009969/3767761230468750000\) | \(13257786322649597167968750000\) | \([2]\) | \(298598400\) | \(4.0981\) | |
288990.ea2 | 288990ea2 | \([1, -1, 1, -631201343, -6103637247769]\) | \(7903870428425797297009/886464000000\) | \(3119239669351104000000\) | \([2]\) | \(99532800\) | \(3.5488\) | |
288990.ea3 | 288990ea1 | \([1, -1, 1, -39349823, -95870838553]\) | \(-1914980734749238129/20440940544000\) | \(-71926432008171945984000\) | \([2]\) | \(49766400\) | \(3.2023\) | \(\Gamma_0(N)\)-optimal |
288990.ea4 | 288990ea3 | \([1, -1, 1, 130028737, -499134322969]\) | \(69096190760262356111/70568821500000000\) | \(-248313600374247661500000000\) | \([2]\) | \(149299200\) | \(3.7516\) |
Rank
sage: E.rank()
The elliptic curves in class 288990.ea have rank \(1\).
Complex multiplication
The elliptic curves in class 288990.ea do not have complex multiplication.Modular form 288990.2.a.ea
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.