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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 30345.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30345.l1 | 30345bf4 | \([1, 0, 0, -17687095, -28632233938]\) | \(25351269426118370449/27551475\) | \(665025628864275\) | \([2]\) | \(884736\) | \(2.5630\) | |
30345.l2 | 30345bf3 | \([1, 0, 0, -1378825, -209447500]\) | \(12010404962647729/6166198828125\) | \(148837049681586328125\) | \([2]\) | \(884736\) | \(2.5630\) | |
30345.l3 | 30345bf2 | \([1, 0, 0, -1105720, -447212713]\) | \(6193921595708449/6452105625\) | \(155738144718725625\) | \([2, 2]\) | \(442368\) | \(2.2164\) | |
30345.l4 | 30345bf1 | \([1, 0, 0, -52315, -10471000]\) | \(-656008386769/1581036975\) | \(-38162389075613775\) | \([4]\) | \(221184\) | \(1.8698\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 30345.l have rank \(1\).
Complex multiplication
The elliptic curves in class 30345.l do not have complex multiplication.Modular form 30345.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.