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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 394944du
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
394944.du4 | 394944du1 | \([0, -1, 0, -51625537, 107426494177]\) | \(32765849647039657/8229948198912\) | \(3822021577595352942379008\) | \([2]\) | \(61931520\) | \(3.4269\) | \(\Gamma_0(N)\)-optimal |
394944.du2 | 394944du2 | \([0, -1, 0, -767790657, 8188204009185]\) | \(107784459654566688937/10704361149504\) | \(4971149064241533614555136\) | \([2, 2]\) | \(123863040\) | \(3.7734\) | |
394944.du1 | 394944du3 | \([0, -1, 0, -12284357697, 524059004685537]\) | \(441453577446719855661097/4354701912\) | \(2022341364658898731008\) | \([2]\) | \(247726080\) | \(4.1200\) | |
394944.du3 | 394944du4 | \([0, -1, 0, -709865537, 9475705651425]\) | \(-85183593440646799657/34223681512621656\) | \(-15893617559255523924381794304\) | \([2]\) | \(247726080\) | \(4.1200\) |
Rank
sage: E.rank()
The elliptic curves in class 394944du have rank \(0\).
Complex multiplication
The elliptic curves in class 394944du do not have complex multiplication.Modular form 394944.2.a.du
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.