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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 12675a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12675.s2 | 12675a1 | \([0, -1, 1, 1244967, 277228343]\) | \(16742875136/12301875\) | \(-156797146303154296875\) | \([]\) | \(359424\) | \(2.5634\) | \(\Gamma_0(N)\)-optimal |
| 12675.s1 | 12675a2 | \([0, -1, 1, -13584783, -23784041032]\) | \(-21752792449024/6591796875\) | \(-84017675273895263671875\) | \([]\) | \(1078272\) | \(3.1127\) |
Rank
sage: E.rank()
The elliptic curves in class 12675a have rank \(1\).
Complex multiplication
The elliptic curves in class 12675a do not have complex multiplication.Modular form 12675.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
