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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 389376.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
389376.e1 | 389376e2 | \([0, 0, 0, -312, 0]\) | \(1728\) | \(1943764992\) | \([2]\) | \(233472\) | \(0.47179\) | \(-4\) | |
389376.e2 | 389376e1 | \([0, 0, 0, 78, 0]\) | \(1728\) | \(-30371328\) | \([2]\) | \(116736\) | \(0.12522\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 389376.e have rank \(0\).
Complex multiplication
Each elliptic curve in class 389376.e has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 389376.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.