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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 45968p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45968.k3 | 45968p1 | \([0, 0, 0, -1859, -13182]\) | \(35937/17\) | \(336100364288\) | \([2]\) | \(36864\) | \(0.90584\) | \(\Gamma_0(N)\)-optimal |
| 45968.k2 | 45968p2 | \([0, 0, 0, -15379, 725010]\) | \(20346417/289\) | \(5713706192896\) | \([2, 2]\) | \(73728\) | \(1.2524\) | |
| 45968.k4 | 45968p3 | \([0, 0, 0, -1859, 1955330]\) | \(-35937/83521\) | \(-1651261089746944\) | \([2]\) | \(147456\) | \(1.5990\) | |
| 45968.k1 | 45968p4 | \([0, 0, 0, -245219, 46738978]\) | \(82483294977/17\) | \(336100364288\) | \([2]\) | \(147456\) | \(1.5990\) |
Rank
sage: E.rank()
The elliptic curves in class 45968p have rank \(1\).
Complex multiplication
The elliptic curves in class 45968p do not have complex multiplication.Modular form 45968.2.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.
