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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 310464ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 310464.ef4 | 310464ef1 | \([0, 0, 0, -2738316, 6377895664]\) | \(-100999381393/723148272\) | \(-16258596129287153123328\) | \([2]\) | \(14155776\) | \(2.9451\) | \(\Gamma_0(N)\)-optimal |
| 310464.ef3 | 310464ef2 | \([0, 0, 0, -71040396, 229944263920]\) | \(1763535241378513/4612311396\) | \(103698938535350086139904\) | \([2, 2]\) | \(28311552\) | \(3.2916\) | |
| 310464.ef1 | 310464ef3 | \([0, 0, 0, -1135931916, 14735896549360]\) | \(7209828390823479793/49509306\) | \(1113121391646349983744\) | \([2]\) | \(56623104\) | \(3.6382\) | |
| 310464.ef2 | 310464ef4 | \([0, 0, 0, -98982156, 32239546864]\) | \(4770223741048753/2740574865798\) | \(61616547574471728708452352\) | \([2]\) | \(56623104\) | \(3.6382\) |
Rank
sage: E.rank()
The elliptic curves in class 310464ef have rank \(2\).
Complex multiplication
The elliptic curves in class 310464ef do not have complex multiplication.Modular form 310464.2.a.ef
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
