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SageMath
E = EllipticCurve("md1")
E.isogeny_class()
Elliptic curves in class 242550md
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
242550.md3 | 242550md1 | \([1, -1, 1, -617630, 3635869997]\) | \(-19443408769/4249907200\) | \(-5695282111780800000000\) | \([2]\) | \(15925248\) | \(2.8538\) | \(\Gamma_0(N)\)-optimal |
242550.md2 | 242550md2 | \([1, -1, 1, -39425630, 94446589997]\) | \(5057359576472449/51765560000\) | \(69370801290511875000000\) | \([2]\) | \(31850496\) | \(3.2003\) | |
242550.md4 | 242550md3 | \([1, -1, 1, 5556370, -97938778003]\) | \(14156681599871/3100231750000\) | \(-4154607053102214843750000\) | \([2]\) | \(47775744\) | \(3.4031\) | |
242550.md1 | 242550md4 | \([1, -1, 1, -287929130, -1827155344003]\) | \(1969902499564819009/63690429687500\) | \(85351267173751831054687500\) | \([2]\) | \(95551488\) | \(3.7496\) |
Rank
sage: E.rank()
The elliptic curves in class 242550md have rank \(1\).
Complex multiplication
The elliptic curves in class 242550md do not have complex multiplication.Modular form 242550.2.a.md
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.