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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3465p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3465.q3 | 3465p1 | \([1, -1, 0, -99, 328]\) | \(148035889/31185\) | \(22733865\) | \([2]\) | \(768\) | \(0.12630\) | \(\Gamma_0(N)\)-optimal |
3465.q2 | 3465p2 | \([1, -1, 0, -504, -3965]\) | \(19443408769/1334025\) | \(972504225\) | \([2, 2]\) | \(1536\) | \(0.47288\) | |
3465.q1 | 3465p3 | \([1, -1, 0, -7929, -269780]\) | \(75627935783569/396165\) | \(288804285\) | \([2]\) | \(3072\) | \(0.81945\) | |
3465.q4 | 3465p4 | \([1, -1, 0, 441, -17762]\) | \(12994449551/192163125\) | \(-140086918125\) | \([4]\) | \(3072\) | \(0.81945\) |
Rank
sage: E.rank()
The elliptic curves in class 3465p have rank \(0\).
Complex multiplication
The elliptic curves in class 3465p do not have complex multiplication.Modular form 3465.2.a.p
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.