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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 87360ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 87360.m4 | 87360ef1 | \([0, -1, 0, -161, -18879]\) | \(-1771561/589680\) | \(-154581073920\) | \([2]\) | \(73728\) | \(0.82617\) | \(\Gamma_0(N)\)-optimal |
| 87360.m3 | 87360ef2 | \([0, -1, 0, -11681, -477375]\) | \(672451615081/7452900\) | \(1953733017600\) | \([2, 2]\) | \(147456\) | \(1.1727\) | |
| 87360.m2 | 87360ef3 | \([0, -1, 0, -21281, 430785]\) | \(4066120948681/2057248830\) | \(539295437291520\) | \([2]\) | \(294912\) | \(1.5193\) | |
| 87360.m1 | 87360ef4 | \([0, -1, 0, -186401, -30913599]\) | \(2732315424539401/341250\) | \(89456640000\) | \([2]\) | \(294912\) | \(1.5193\) |
Rank
sage: E.rank()
The elliptic curves in class 87360ef have rank \(1\).
Complex multiplication
The elliptic curves in class 87360ef do not have complex multiplication.Modular form 87360.2.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.
