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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 56350.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 56350.k1 | 56350n4 | \([1, -1, 0, -10518692, 13133415966]\) | \(70016546394529281/1610\) | \(2959607656250\) | \([2]\) | \(1179648\) | \(2.3676\) | |
| 56350.k2 | 56350n2 | \([1, -1, 0, -657442, 205317216]\) | \(17095749786081/2592100\) | \(4764968326562500\) | \([2, 2]\) | \(589824\) | \(2.0210\) | |
| 56350.k3 | 56350n3 | \([1, -1, 0, -596192, 245068466]\) | \(-12748946194881/6718982410\) | \(-12351274399282656250\) | \([2]\) | \(1179648\) | \(2.3676\) | |
| 56350.k4 | 56350n1 | \([1, -1, 0, -44942, 2579716]\) | \(5461074081/1610000\) | \(2959607656250000\) | \([2]\) | \(294912\) | \(1.6744\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 56350.k have rank \(1\).
Complex multiplication
The elliptic curves in class 56350.k do not have complex multiplication.Modular form 56350.2.a.k
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 LMFDB 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 LMFDB labels.
