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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 325325t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
325325.t4 | 325325t1 | \([1, -1, 1, -66280, -55363278]\) | \(-426957777/17320303\) | \(-1306278037548859375\) | \([2]\) | \(3268608\) | \(2.1563\) | \(\Gamma_0(N)\)-optimal |
325325.t3 | 325325t2 | \([1, -1, 1, -2622405, -1624824028]\) | \(26444947540257/169338169\) | \(12771296846448765625\) | \([2, 2]\) | \(6537216\) | \(2.5028\) | |
325325.t2 | 325325t3 | \([1, -1, 1, -4249030, 632931472]\) | \(112489728522417/62811265517\) | \(4737155964044457078125\) | \([2]\) | \(13074432\) | \(2.8494\) | |
325325.t1 | 325325t4 | \([1, -1, 1, -41893780, -104358741028]\) | \(107818231938348177/4463459\) | \(336629126130171875\) | \([2]\) | \(13074432\) | \(2.8494\) |
Rank
sage: E.rank()
The elliptic curves in class 325325t have rank \(1\).
Complex multiplication
The elliptic curves in class 325325t do not have complex multiplication.Modular form 325325.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.