Show commands:
SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 115920es
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 115920.es4 | 115920es1 | \([0, 0, 0, -164307, 76750994]\) | \(-164287467238609/757170892800\) | \(-2260900171166515200\) | \([2]\) | \(1769472\) | \(2.2066\) | \(\Gamma_0(N)\)-optimal |
| 115920.es3 | 115920es2 | \([0, 0, 0, -3896787, 2955986066]\) | \(2191574502231419089/4115217960000\) | \(12287974985072640000\) | \([2, 2]\) | \(3538944\) | \(2.5532\) | |
| 115920.es2 | 115920es3 | \([0, 0, 0, -5192787, 820437266]\) | \(5186062692284555089/2903809817953800\) | \(8670729655452959539200\) | \([2]\) | \(7077888\) | \(2.8997\) | |
| 115920.es1 | 115920es4 | \([0, 0, 0, -62320467, 189362579474]\) | \(8964546681033941529169/31696875000\) | \(94646361600000000\) | \([2]\) | \(7077888\) | \(2.8997\) |
Rank
sage: E.rank()
The elliptic curves in class 115920es have rank \(1\).
Complex multiplication
The elliptic curves in class 115920es do not have complex multiplication.Modular form 115920.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
