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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 12138a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12138.d3 | 12138a1 | \([1, 1, 0, -156499, 24133165]\) | \(-3574558889/64512\) | \(-7650341088574464\) | \([2]\) | \(108800\) | \(1.8441\) | \(\Gamma_0(N)\)-optimal |
12138.d2 | 12138a2 | \([1, 1, 0, -2514739, 1533878413]\) | \(14830727012009/4704\) | \(557837371041888\) | \([2]\) | \(217600\) | \(2.1907\) | |
12138.d4 | 12138a3 | \([1, 1, 0, 924361, -985881375]\) | \(736558976791/3969746172\) | \(-470763768769574519484\) | \([2]\) | \(544000\) | \(2.6488\) | |
12138.d1 | 12138a4 | \([1, 1, 0, -11014229, -12664210113]\) | \(1246079601667529/137282971014\) | \(16280096011749462857958\) | \([2]\) | \(1088000\) | \(2.9954\) |
Rank
sage: E.rank()
The elliptic curves in class 12138a have rank \(1\).
Complex multiplication
The elliptic curves in class 12138a do not have complex multiplication.Modular form 12138.2.a.a
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.