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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 8349c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8349.d6 | 8349c1 | \([1, 0, 1, 3748, 259301]\) | \(3288008303/18259263\) | \(-32347398219543\) | \([2]\) | \(15360\) | \(1.2743\) | \(\Gamma_0(N)\)-optimal |
8349.d5 | 8349c2 | \([1, 0, 1, -45257, 3336815]\) | \(5786435182177/627352209\) | \(1111392706728249\) | \([2, 2]\) | \(30720\) | \(1.6209\) | |
8349.d4 | 8349c3 | \([1, 0, 1, -170492, -23513569]\) | \(309368403125137/44372288367\) | \(78608215551730887\) | \([2]\) | \(61440\) | \(1.9675\) | |
8349.d2 | 8349c4 | \([1, 0, 1, -704102, 227344115]\) | \(21790813729717297/304746849\) | \(539877632561289\) | \([2, 2]\) | \(61440\) | \(1.9675\) | |
8349.d1 | 8349c5 | \([1, 0, 1, -11265587, 14552942369]\) | \(89254274298475942657/17457\) | \(30926140377\) | \([2]\) | \(122880\) | \(2.3141\) | |
8349.d3 | 8349c6 | \([1, 0, 1, -684137, 240848441]\) | \(-19989223566735457/2584262514273\) | \(-4578178684047990153\) | \([2]\) | \(122880\) | \(2.3141\) |
Rank
sage: E.rank()
The elliptic curves in class 8349c have rank \(1\).
Complex multiplication
The elliptic curves in class 8349c do not have complex multiplication.Modular form 8349.2.a.c
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.