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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 104040i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
104040.q3 | 104040i1 | \([0, 0, 0, -371943, -87284358]\) | \(1263257424/425\) | \(1914476112748800\) | \([2]\) | \(589824\) | \(1.9046\) | \(\Gamma_0(N)\)-optimal |
104040.q2 | 104040i2 | \([0, 0, 0, -423963, -61284762]\) | \(467720676/180625\) | \(3254609391672960000\) | \([2, 2]\) | \(1179648\) | \(2.2512\) | |
104040.q4 | 104040i3 | \([0, 0, 0, 1344717, -440136018]\) | \(7462174302/6640625\) | \(-239309514093600000000\) | \([2]\) | \(2359296\) | \(2.5978\) | |
104040.q1 | 104040i4 | \([0, 0, 0, -3024963, 1981540638]\) | \(84944038338/2088025\) | \(75246569135478835200\) | \([2]\) | \(2359296\) | \(2.5978\) |
Rank
sage: E.rank()
The elliptic curves in class 104040i have rank \(0\).
Complex multiplication
The elliptic curves in class 104040i do not have complex multiplication.Modular form 104040.2.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 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.