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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 68970a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68970.g2 | 68970a1 | \([1, 1, 0, -2314248, -1272192192]\) | \(581325709271579/40343961600\) | \(95128951100512665600\) | \([2]\) | \(2703360\) | \(2.5814\) | \(\Gamma_0(N)\)-optimal |
68970.g1 | 68970a2 | \([1, 1, 0, -36387848, -84500367552]\) | \(2259741076336189979/12126827520\) | \(28594424949939256320\) | \([2]\) | \(5406720\) | \(2.9279\) |
Rank
sage: E.rank()
The elliptic curves in class 68970a have rank \(1\).
Complex multiplication
The elliptic curves in class 68970a do not have complex multiplication.Modular form 68970.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.