Show commands:
SageMath
E = EllipticCurve("hk1")
E.isogeny_class()
Elliptic curves in class 394944.hk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 394944.hk1 | 394944hk3 | \([0, 1, 0, -128508334753, -17731525731909409]\) | \(505384091400037554067434625/815656731648\) | \(378794319590211591340032\) | \([2]\) | \(796262400\) | \(4.6796\) | |
| 394944.hk2 | 394944hk4 | \([0, 1, 0, -128507095713, -17731884750935841]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-9428898214226683510900131889152\) | \([2]\) | \(1592524800\) | \(5.0262\) | |
| 394944.hk3 | 394944hk1 | \([0, 1, 0, -1590989473, -24179701612321]\) | \(959024269496848362625/11151660319506432\) | \(5178876626805754214574194688\) | \([2]\) | \(265420800\) | \(4.1303\) | \(\Gamma_0(N)\)-optimal |
| 394944.hk4 | 394944hk2 | \([0, 1, 0, -322212513, -61681449729825]\) | \(-7966267523043306625/3534510366354604032\) | \(-1641441058915491164594884313088\) | \([2]\) | \(530841600\) | \(4.4769\) |
Rank
sage: E.rank()
The elliptic curves in class 394944.hk have rank \(1\).
Complex multiplication
The elliptic curves in class 394944.hk do not have complex multiplication.Modular form 394944.2.a.hk
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.
