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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 212415y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212415.m4 | 212415y1 | \([1, 1, 1, -2563436, 3588989564]\) | \(-656008386769/1581036975\) | \(-4489766912356885014975\) | \([2]\) | \(10616832\) | \(2.8428\) | \(\Gamma_0(N)\)-optimal |
212415.m3 | 212415y2 | \([1, 1, 1, -54180281, 153339780278]\) | \(6193921595708449/6452105625\) | \(18322436988013351055625\) | \([2, 2]\) | \(21233664\) | \(3.1894\) | |
212415.m1 | 212415y3 | \([1, 1, 1, -866667656, 9819989573078]\) | \(25351269426118370449/27551475\) | \(78239600210253089475\) | \([2]\) | \(42467328\) | \(3.5359\) | |
212415.m2 | 212415y4 | \([1, 1, 1, -67562426, 71772930074]\) | \(12010404962647729/6166198828125\) | \(17510530057988949917578125\) | \([2]\) | \(42467328\) | \(3.5359\) |
Rank
sage: E.rank()
The elliptic curves in class 212415y have rank \(0\).
Complex multiplication
The elliptic curves in class 212415y do not have complex multiplication.Modular form 212415.2.a.y
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.