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SageMath
E = EllipticCurve("ht1")
E.isogeny_class()
Elliptic curves in class 388080ht
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.ht4 | 388080ht1 | \([0, 0, 0, -22279467, -2765779814]\) | \(3481467828171481/2005331497785\) | \(704469007909435563970560\) | \([2]\) | \(47185920\) | \(3.2655\) | \(\Gamma_0(N)\)-optimal |
| 388080.ht2 | 388080ht2 | \([0, 0, 0, -253751547, -1552101000086]\) | \(5143681768032498601/14238434358225\) | \(5001933963338066719641600\) | \([2, 2]\) | \(94371840\) | \(3.6121\) | |
| 388080.ht3 | 388080ht3 | \([0, 0, 0, -153732747, -2788113326726]\) | \(-1143792273008057401/8897444448004035\) | \(-3125654720996525155525570560\) | \([2]\) | \(188743680\) | \(3.9587\) | |
| 388080.ht1 | 388080ht4 | \([0, 0, 0, -4057323627, -99473542770854]\) | \(21026497979043461623321/161783881875\) | \(56834359249882959360000\) | \([2]\) | \(188743680\) | \(3.9587\) |
Rank
sage: E.rank()
The elliptic curves in class 388080ht have rank \(1\).
Complex multiplication
The elliptic curves in class 388080ht do not have complex multiplication.Modular form 388080.2.a.ht
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.
