Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 19481.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19481.e1 | 19481b3 | \([1, -1, 0, -14966, -700815]\) | \(209267191953/55223\) | \(97830913103\) | \([2]\) | \(25600\) | \(1.0933\) | |
19481.e2 | 19481b2 | \([1, -1, 0, -1051, -7848]\) | \(72511713/25921\) | \(45920632681\) | \([2, 2]\) | \(12800\) | \(0.74674\) | |
19481.e3 | 19481b1 | \([1, -1, 0, -446, 3647]\) | \(5545233/161\) | \(285221321\) | \([2]\) | \(6400\) | \(0.40017\) | \(\Gamma_0(N)\)-optimal |
19481.e4 | 19481b4 | \([1, -1, 0, 3184, -57821]\) | \(2014698447/1958887\) | \(-3470287812607\) | \([2]\) | \(25600\) | \(1.0933\) |
Rank
sage: E.rank()
The elliptic curves in class 19481.e have rank \(0\).
Complex multiplication
The elliptic curves in class 19481.e do not have complex multiplication.Modular form 19481.2.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 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.