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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 43758.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43758.i1 | 43758h4 | \([1, -1, 0, -22586878296, 1306574318542144]\) | \(1748094148784980747354970849498497/887694600425282263291392\) | \(647129363710030769939424768\) | \([2]\) | \(75644928\) | \(4.4773\) | |
43758.i2 | 43758h3 | \([1, -1, 0, -3089739096, -36353344521920]\) | \(4474676144192042711273397261697/1806328356954994499451382272\) | \(1316813372220190990100057676288\) | \([2]\) | \(75644928\) | \(4.4773\) | |
43758.i3 | 43758h2 | \([1, -1, 0, -1419316056, 20183459605312]\) | \(433744050935826360922067531137/9612122270219882316693504\) | \(7007237134990294208869564416\) | \([2, 2]\) | \(37822464\) | \(4.1307\) | |
43758.i4 | 43758h1 | \([1, -1, 0, 8058024, 966722366272]\) | \(79374649975090937760383/553856914190911653543936\) | \(-403761690445174595433529344\) | \([2]\) | \(18911232\) | \(3.7842\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 43758.i have rank \(0\).
Complex multiplication
The elliptic curves in class 43758.i do not have complex multiplication.Modular form 43758.2.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.