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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 13475.t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13475.t1 | 13475h3 | \([1, -1, 0, -12576692, -17163988409]\) | \(119678115308998401/1925\) | \(3538661328125\) | \([2]\) | \(294912\) | \(2.4066\) | |
| 13475.t2 | 13475h4 | \([1, -1, 0, -853442, -219310659]\) | \(37397086385121/10316796875\) | \(18965013055419921875\) | \([2]\) | \(294912\) | \(2.4066\) | |
| 13475.t3 | 13475h2 | \([1, -1, 0, -786067, -268022784]\) | \(29220958012401/3705625\) | \(6811923056640625\) | \([2, 2]\) | \(147456\) | \(2.0600\) | |
| 13475.t4 | 13475h1 | \([1, -1, 0, -44942, -4923409]\) | \(-5461074081/2562175\) | \(-4709958227734375\) | \([2]\) | \(73728\) | \(1.7134\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13475.t have rank \(1\).
Complex multiplication
The elliptic curves in class 13475.t do not have complex multiplication.Modular form 13475.2.a.t
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.
