Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3006e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3006.d2 | 3006e1 | \([1, -1, 1, -41, 425]\) | \(-10218313/96192\) | \(-70123968\) | \([2]\) | \(1152\) | \(0.18788\) | \(\Gamma_0(N)\)-optimal |
3006.d1 | 3006e2 | \([1, -1, 1, -1121, 14681]\) | \(213525509833/669336\) | \(487945944\) | \([2]\) | \(2304\) | \(0.53445\) |
Rank
sage: E.rank()
The elliptic curves in class 3006e have rank \(1\).
Complex multiplication
The elliptic curves in class 3006e do not have complex multiplication.Modular form 3006.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.