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SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 47040em
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.bm3 | 47040em1 | \([0, -1, 0, -996, 12426]\) | \(14526784/15\) | \(112943040\) | \([2]\) | \(24576\) | \(0.46311\) | \(\Gamma_0(N)\)-optimal |
| 47040.bm2 | 47040em2 | \([0, -1, 0, -1241, 6105]\) | \(438976/225\) | \(108425318400\) | \([2, 2]\) | \(49152\) | \(0.80969\) | |
| 47040.bm4 | 47040em3 | \([0, -1, 0, 4639, 42561]\) | \(2863288/1875\) | \(-7228354560000\) | \([2]\) | \(98304\) | \(1.1563\) | |
| 47040.bm1 | 47040em4 | \([0, -1, 0, -11041, -438815]\) | \(38614472/405\) | \(1561324584960\) | \([2]\) | \(98304\) | \(1.1563\) |
Rank
sage: E.rank()
The elliptic curves in class 47040em have rank \(1\).
Complex multiplication
The elliptic curves in class 47040em do not have complex multiplication.Modular form 47040.2.a.em
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.
