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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 45675.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45675.e1 | 45675k4 | \([1, -1, 1, -46040405, -120230661778]\) | \(947531277805646290177/38367\) | \(437024109375\) | \([2]\) | \(1572864\) | \(2.6447\) | |
| 45675.e2 | 45675k6 | \([1, -1, 1, -9555530, 9244638722]\) | \(8471112631466271697/1662662681263647\) | \(18938767103768729109375\) | \([2]\) | \(3145728\) | \(2.9913\) | |
| 45675.e3 | 45675k3 | \([1, -1, 1, -2932655, -1802316778]\) | \(244883173420511137/18418027974129\) | \(209792849892813140625\) | \([2, 2]\) | \(1572864\) | \(2.6447\) | |
| 45675.e4 | 45675k2 | \([1, -1, 1, -2877530, -1878058528]\) | \(231331938231569617/1472026689\) | \(16767304004390625\) | \([2, 2]\) | \(786432\) | \(2.2981\) | |
| 45675.e5 | 45675k1 | \([1, -1, 1, -176405, -30489028]\) | \(-53297461115137/4513839183\) | \(-51415449443859375\) | \([2]\) | \(393216\) | \(1.9515\) | \(\Gamma_0(N)\)-optimal |
| 45675.e6 | 45675k5 | \([1, -1, 1, 2808220, -8002461778]\) | \(215015459663151503/2552757445339983\) | \(-29077502775825743859375\) | \([2]\) | \(3145728\) | \(2.9913\) |
Rank
sage: E.rank()
The elliptic curves in class 45675.e have rank \(0\).
Complex multiplication
The elliptic curves in class 45675.e do not have complex multiplication.Modular form 45675.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
