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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 13328p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13328.k3 | 13328p1 | \([0, 0, 0, -14651, 671594]\) | \(721734273/13328\) | \(6422633971712\) | \([2]\) | \(18432\) | \(1.2521\) | \(\Gamma_0(N)\)-optimal |
13328.k2 | 13328p2 | \([0, 0, 0, -30331, -1018710]\) | \(6403769793/2775556\) | \(1337513524609024\) | \([2, 2]\) | \(36864\) | \(1.5987\) | |
13328.k1 | 13328p3 | \([0, 0, 0, -414491, -102667446]\) | \(16342588257633/8185058\) | \(3944300087877632\) | \([2]\) | \(73728\) | \(1.9452\) | |
13328.k4 | 13328p4 | \([0, 0, 0, 102949, -7549430]\) | \(250404380127/196003234\) | \(-94452058017243136\) | \([2]\) | \(73728\) | \(1.9452\) |
Rank
sage: E.rank()
The elliptic curves in class 13328p have rank \(1\).
Complex multiplication
The elliptic curves in class 13328p do not have complex multiplication.Modular form 13328.2.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.