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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 10005.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10005.l1 | 10005g2 | \([1, 1, 0, -112452, -14561001]\) | \(157264717208387436361/4368589453125\) | \(4368589453125\) | \([2]\) | \(66048\) | \(1.5282\) | |
10005.l2 | 10005g1 | \([1, 1, 0, -6747, -248544]\) | \(-33974761330806841/6424789539375\) | \(-6424789539375\) | \([2]\) | \(33024\) | \(1.1817\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10005.l have rank \(1\).
Complex multiplication
The elliptic curves in class 10005.l do not have complex multiplication.Modular form 10005.2.a.l
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.