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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 350d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 350.f5 | 350d1 | \([1, 1, 1, -13, 31]\) | \(-15625/28\) | \(-437500\) | \([2]\) | \(48\) | \(-0.22737\) | \(\Gamma_0(N)\)-optimal |
| 350.f4 | 350d2 | \([1, 1, 1, -263, 1531]\) | \(128787625/98\) | \(1531250\) | \([2]\) | \(96\) | \(0.11921\) | |
| 350.f6 | 350d3 | \([1, 1, 1, 112, -719]\) | \(9938375/21952\) | \(-343000000\) | \([2]\) | \(144\) | \(0.32194\) | |
| 350.f3 | 350d4 | \([1, 1, 1, -888, -8719]\) | \(4956477625/941192\) | \(14706125000\) | \([2]\) | \(288\) | \(0.66851\) | |
| 350.f2 | 350d5 | \([1, 1, 1, -4263, -109219]\) | \(-548347731625/1835008\) | \(-28672000000\) | \([2]\) | \(432\) | \(0.87125\) | |
| 350.f1 | 350d6 | \([1, 1, 1, -68263, -6893219]\) | \(2251439055699625/25088\) | \(392000000\) | \([2]\) | \(864\) | \(1.2178\) |
Rank
sage: E.rank()
The elliptic curves in class 350d have rank \(0\).
Complex multiplication
The elliptic curves in class 350d do not have complex multiplication.Modular form 350.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
