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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 25230e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25230.d4 | 25230e1 | \([1, 1, 0, -3767697, -616823019]\) | \(9944061759313921/5479747200000\) | \(3259481427744451200000\) | \([2]\) | \(1344000\) | \(2.8186\) | \(\Gamma_0(N)\)-optimal |
| 25230.d3 | 25230e2 | \([1, 1, 0, -36465777, 84221615349]\) | \(9015548596898711041/63863437500000\) | \(37987461984225937500000\) | \([2]\) | \(2688000\) | \(3.1651\) | |
| 25230.d2 | 25230e3 | \([1, 1, 0, -1796863797, -29317838120199]\) | \(1078651622544688278688321/3692006820\) | \(2196091757827049220\) | \([2]\) | \(6720000\) | \(3.6233\) | |
| 25230.d1 | 25230e4 | \([1, 1, 0, -1796889027, -29316973674801]\) | \(1078697059648930939019041/63106084995030150\) | \(37536971052052102318128150\) | \([2]\) | \(13440000\) | \(3.9699\) |
Rank
sage: E.rank()
The elliptic curves in class 25230e have rank \(0\).
Complex multiplication
The elliptic curves in class 25230e do not have complex multiplication.Modular form 25230.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
