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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 433200cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
433200.cn2 | 433200cn1 | \([0, -1, 0, 77106592, -770894930688]\) | \(129205871/729000\) | \(-286051027323963456000000000\) | \([]\) | \(141834240\) | \(3.7585\) | \(\Gamma_0(N)\)-optimal |
433200.cn1 | 433200cn2 | \([0, -1, 0, -4614449408, -120780897410688]\) | \(-27692833539889/35156250\) | \(-13794899080052250000000000000\) | \([]\) | \(425502720\) | \(4.3078\) |
Rank
sage: E.rank()
The elliptic curves in class 433200cn have rank \(1\).
Complex multiplication
The elliptic curves in class 433200cn do not have complex multiplication.Modular form 433200.2.a.cn
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.