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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 67158t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67158.d4 | 67158t1 | \([1, -1, 0, -125433, 19157229]\) | \(-299387428352690833/43513123110912\) | \(-31721066747854848\) | \([2]\) | \(688128\) | \(1.8967\) | \(\Gamma_0(N)\)-optimal |
| 67158.d3 | 67158t2 | \([1, -1, 0, -2072313, 1148737005]\) | \(1350088866691276036753/23380861061376\) | \(17044647713743104\) | \([2, 2]\) | \(1376256\) | \(2.2433\) | |
| 67158.d2 | 67158t3 | \([1, -1, 0, -2137833, 1072275165]\) | \(1482236924759943084433/177107469272815536\) | \(129111345099882525744\) | \([2]\) | \(2752512\) | \(2.5899\) | |
| 67158.d1 | 67158t4 | \([1, -1, 0, -33156873, 73494941949]\) | \(5529895044677685547285393/1658533968\) | \(1209071262672\) | \([2]\) | \(2752512\) | \(2.5899\) |
Rank
sage: E.rank()
The elliptic curves in class 67158t have rank \(0\).
Complex multiplication
The elliptic curves in class 67158t do not have complex multiplication.Modular form 67158.2.a.t
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.
