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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 189618.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 189618.i1 | 189618bs4 | \([1, 1, 0, -424131381339, -106316153461934307]\) | \(1748094148784980747354970849498497/887694600425282263291392\) | \(4284732286584156255995260528128\) | \([2]\) | \(1588543488\) | \(5.2105\) | |
| 189618.i2 | 189618bs3 | \([1, 1, 0, -58018434139, 2958123786906397]\) | \(4474676144192042711273397261697/1806328356954994499451382272\) | \(8718801970305580044902427012930048\) | \([2]\) | \(1588543488\) | \(5.2105\) | |
| 189618.i3 | 189618bs2 | \([1, 1, 0, -26651601499, -1642317815705315]\) | \(433744050935826360922067531137/9612122270219882316693504\) | \(46395878282997759945157055348736\) | \([2, 2]\) | \(794271744\) | \(4.8639\) | |
| 189618.i4 | 189618bs1 | \([1, 1, 0, 151311781, -78662657863395]\) | \(79374649975090937760383/553856914190911653543936\) | \(-2673361538128920087530752180224\) | \([2]\) | \(397135872\) | \(4.5173\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 189618.i have rank \(1\).
Complex multiplication
The elliptic curves in class 189618.i do not have complex multiplication.Modular form 189618.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
