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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 11025bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11025.bn2 | 11025bc1 | \([0, 0, 1, 735, 33871]\) | \(20480/243\) | \(-521029185075\) | \([]\) | \(15840\) | \(0.92916\) | \(\Gamma_0(N)\)-optimal |
11025.bn1 | 11025bc2 | \([0, 0, 1, -91875, -10986719]\) | \(-102400/3\) | \(-2512679326171875\) | \([]\) | \(79200\) | \(1.7339\) |
Rank
sage: E.rank()
The elliptic curves in class 11025bc have rank \(1\).
Complex multiplication
The elliptic curves in class 11025bc do not have complex multiplication.Modular form 11025.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.