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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 139230i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139230.eo4 | 139230i1 | \([1, -1, 1, -56215112, -552280207989]\) | \(-26949791983733109138764089/165161952797784563712000\) | \(-120403063589584946946048000\) | \([4]\) | \(51904512\) | \(3.6881\) | \(\Gamma_0(N)\)-optimal |
139230.eo3 | 139230i2 | \([1, -1, 1, -1411531592, -20370259905141]\) | \(426646307804307769001905914169/998470877001641316000000\) | \(727885269334196519364000000\) | \([2, 2]\) | \(103809024\) | \(4.0346\) | |
139230.eo2 | 139230i3 | \([1, -1, 1, -1936321592, -3851550033141]\) | \(1101358349464662961278219354169/628567168199833707765102000\) | \(458225465617678772960759358000\) | \([2]\) | \(207618048\) | \(4.3812\) | |
139230.eo1 | 139230i4 | \([1, -1, 1, -22571805272, -1305255934192629]\) | \(1744596788171434949302427839201849/9588363813082031250000\) | \(6989917219736800781250000\) | \([2]\) | \(207618048\) | \(4.3812\) |
Rank
sage: E.rank()
The elliptic curves in class 139230i have rank \(0\).
Complex multiplication
The elliptic curves in class 139230i do not have complex multiplication.Modular form 139230.2.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.