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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 27378.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27378.c1 | 27378i2 | \([1, -1, 0, -9411, -381367]\) | \(-35937/4\) | \(-10260656807076\) | \([]\) | \(84240\) | \(1.2340\) | |
27378.c2 | 27378i1 | \([1, -1, 0, 729, 573]\) | \(109503/64\) | \(-25022177856\) | \([]\) | \(28080\) | \(0.68467\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27378.c have rank \(1\).
Complex multiplication
The elliptic curves in class 27378.c do not have complex multiplication.Modular form 27378.2.a.c
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.