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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 19074.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19074.a1 | 19074b3 | \([1, 1, 0, -23270, -1370796]\) | \(57736239625/255552\) | \(6168404033088\) | \([2]\) | \(55296\) | \(1.3065\) | |
| 19074.a2 | 19074b4 | \([1, 1, 0, -11710, -2718692]\) | \(-7357983625/127552392\) | \(-3078804663015048\) | \([2]\) | \(110592\) | \(1.6531\) | |
| 19074.a3 | 19074b1 | \([1, 1, 0, -1595, 22473]\) | \(18609625/1188\) | \(28675431972\) | \([2]\) | \(18432\) | \(0.75722\) | \(\Gamma_0(N)\)-optimal |
| 19074.a4 | 19074b2 | \([1, 1, 0, 1295, 98191]\) | \(9938375/176418\) | \(-4258301647842\) | \([2]\) | \(36864\) | \(1.1038\) |
Rank
sage: E.rank()
The elliptic curves in class 19074.a have rank \(2\).
Complex multiplication
The elliptic curves in class 19074.a do not have complex multiplication.Modular form 19074.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
