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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 439824es
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 439824.es3 | 439824es1 | \([0, 1, 0, -161071248, -778928197356]\) | \(959024269496848362625/11151660319506432\) | \(5373876981471691646435328\) | \([2]\) | \(99532800\) | \(3.5577\) | \(\Gamma_0(N)\)-optimal |
| 439824.es4 | 439824es2 | \([0, 1, 0, -32620688, -1986928643820]\) | \(-7966267523043306625/3534510366354604032\) | \(-1703246274933771508780105728\) | \([2]\) | \(199065600\) | \(3.9043\) | |
| 439824.es1 | 439824es3 | \([0, 1, 0, -13010141328, -571181894307564]\) | \(505384091400037554067434625/815656731648\) | \(393057070373501140992\) | \([2]\) | \(298598400\) | \(4.1070\) | |
| 439824.es2 | 439824es4 | \([0, 1, 0, -13010015888, -571193459223276]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-9783924724486934548594163712\) | \([2]\) | \(597196800\) | \(4.4536\) |
Rank
sage: E.rank()
The elliptic curves in class 439824es have rank \(1\).
Complex multiplication
The elliptic curves in class 439824es do not have complex multiplication.Modular form 439824.2.a.es
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
