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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 16830t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16830.l4 | 16830t1 | \([1, -1, 0, -65790, -6478700]\) | \(43199583152847841/89760000\) | \(65435040000\) | \([2]\) | \(49152\) | \(1.3248\) | \(\Gamma_0(N)\)-optimal |
| 16830.l3 | 16830t2 | \([1, -1, 0, -66510, -6329084]\) | \(44633474953947361/1967006250000\) | \(1433947556250000\) | \([2, 2]\) | \(98304\) | \(1.6713\) | |
| 16830.l2 | 16830t3 | \([1, -1, 0, -179010, 20828416]\) | \(870220733067747361/247623269602500\) | \(180517363540222500\) | \([2, 2]\) | \(196608\) | \(2.0179\) | |
| 16830.l5 | 16830t4 | \([1, -1, 0, 34470, -23919800]\) | \(6213165856218719/342407226562500\) | \(-249614868164062500\) | \([2]\) | \(196608\) | \(2.0179\) | |
| 16830.l1 | 16830t5 | \([1, -1, 0, -2629260, 1641423766]\) | \(2757381641970898311361/379829992662450\) | \(276896064650926050\) | \([2]\) | \(393216\) | \(2.3645\) | |
| 16830.l6 | 16830t6 | \([1, -1, 0, 471240, 136963066]\) | \(15875306080318016639/20322604533582450\) | \(-14815178704981606050\) | \([2]\) | \(393216\) | \(2.3645\) |
Rank
sage: E.rank()
The elliptic curves in class 16830t have rank \(1\).
Complex multiplication
The elliptic curves in class 16830t do not have complex multiplication.Modular form 16830.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
