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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 188598.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
188598.l1 | 188598k3 | \([1, 1, 1, -1387713, 475128639]\) | \(46753267515625/11591221248\) | \(73272317689890865152\) | \([2]\) | \(5806080\) | \(2.5220\) | |
188598.l2 | 188598k1 | \([1, 1, 1, -472458, -125145513]\) | \(1845026709625/793152\) | \(5013801745040448\) | \([2]\) | \(1935360\) | \(1.9727\) | \(\Gamma_0(N)\)-optimal |
188598.l3 | 188598k2 | \([1, 1, 1, -398498, -165557257]\) | \(-1107111813625/1228691592\) | \(-7767005628285784008\) | \([2]\) | \(3870720\) | \(2.3193\) | |
188598.l4 | 188598k4 | \([1, 1, 1, 3345727, 3017932607]\) | \(655215969476375/1001033261568\) | \(-6327894670495907000832\) | \([2]\) | \(11612160\) | \(2.8686\) |
Rank
sage: E.rank()
The elliptic curves in class 188598.l have rank \(1\).
Complex multiplication
The elliptic curves in class 188598.l do not have complex multiplication.Modular form 188598.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.