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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 8085.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8085.t1 | 8085l4 | \([1, 1, 0, -100622, 12243519]\) | \(957681397954009/31185\) | \(3668884065\) | \([2]\) | \(18432\) | \(1.3365\) | |
8085.t2 | 8085l3 | \([1, 1, 0, -9972, -61851]\) | \(932288503609/527295615\) | \(62035801809135\) | \([2]\) | \(18432\) | \(1.3365\) | |
8085.t3 | 8085l2 | \([1, 1, 0, -6297, 188784]\) | \(234770924809/1334025\) | \(156946707225\) | \([2, 2]\) | \(9216\) | \(0.98997\) | |
8085.t4 | 8085l1 | \([1, 1, 0, -172, 6259]\) | \(-4826809/144375\) | \(-16985574375\) | \([2]\) | \(4608\) | \(0.64340\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8085.t have rank \(0\).
Complex multiplication
The elliptic curves in class 8085.t do not have complex multiplication.Modular form 8085.2.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.