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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 78400.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78400.bm1 | 78400m2 | \([0, 1, 0, -12428033, -29287627937]\) | \(-8990558521/10485760\) | \(-247596317629480960000000\) | \([]\) | \(8128512\) | \(3.1829\) | |
78400.bm2 | 78400m1 | \([0, 1, 0, 1291967, 745452063]\) | \(10100279/16000\) | \(-377801998336000000000\) | \([]\) | \(2709504\) | \(2.6336\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 78400.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 78400.bm do not have complex multiplication.Modular form 78400.2.a.bm
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.