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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 42042.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42042.i1 | 42042h4 | \([1, 1, 0, -3193600, 2195132416]\) | \(30618029936661765625/3678951124992\) | \(432824920904183808\) | \([2]\) | \(995328\) | \(2.4088\) | |
42042.i2 | 42042h3 | \([1, 1, 0, -183040, 40173568]\) | \(-5764706497797625/2612665516032\) | \(-307377485295648768\) | \([2]\) | \(497664\) | \(2.0622\) | |
42042.i3 | 42042h2 | \([1, 1, 0, -88225, -5783483]\) | \(645532578015625/252306960048\) | \(29683661542687152\) | \([2]\) | \(331776\) | \(1.8595\) | |
42042.i4 | 42042h1 | \([1, 1, 0, 17615, -639659]\) | \(5137417856375/4510142208\) | \(-530613720628992\) | \([2]\) | \(165888\) | \(1.5129\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42042.i have rank \(0\).
Complex multiplication
The elliptic curves in class 42042.i do not have complex multiplication.Modular form 42042.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.