Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 344760k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 344760.k3 | 344760k1 | \([0, -1, 0, -387911, 93121560]\) | \(83587439220736/6885\) | \(531721279440\) | \([2]\) | \(1769472\) | \(1.6935\) | \(\Gamma_0(N)\)-optimal |
| 344760.k2 | 344760k2 | \([0, -1, 0, -388756, 92696356]\) | \(5258429611216/47403225\) | \(58574416143110400\) | \([2, 2]\) | \(3538944\) | \(2.0401\) | |
| 344760.k4 | 344760k3 | \([0, -1, 0, -114976, 220496860]\) | \(-34008619684/4228250625\) | \(-20898773167109760000\) | \([2]\) | \(7077888\) | \(2.3866\) | |
| 344760.k1 | 344760k4 | \([0, -1, 0, -676056, -62330724]\) | \(6913728144004/3658971285\) | \(18085023261879874560\) | \([2]\) | \(7077888\) | \(2.3866\) |
Rank
sage: E.rank()
The elliptic curves in class 344760k have rank \(1\).
Complex multiplication
The elliptic curves in class 344760k do not have complex multiplication.Modular form 344760.2.a.k
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.
