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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 82368en
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82368.cy4 | 82368en1 | \([0, 0, 0, 207060, 26076112]\) | \(5137417856375/4510142208\) | \(-861901598132011008\) | \([2]\) | \(884736\) | \(2.1290\) | \(\Gamma_0(N)\)-optimal |
| 82368.cy3 | 82368en2 | \([0, 0, 0, -1037100, 231611344]\) | \(645532578015625/252306960048\) | \(48216610930685902848\) | \([2]\) | \(1769472\) | \(2.4755\) | |
| 82368.cy2 | 82368en3 | \([0, 0, 0, -2151660, -1622197424]\) | \(-5764706497797625/2612665516032\) | \(-499288155406290911232\) | \([2]\) | \(2654208\) | \(2.6783\) | |
| 82368.cy1 | 82368en4 | \([0, 0, 0, -37541100, -88524506288]\) | \(30618029936661765625/3678951124992\) | \(703058508544519176192\) | \([2]\) | \(5308416\) | \(3.0249\) |
Rank
sage: E.rank()
The elliptic curves in class 82368en have rank \(0\).
Complex multiplication
The elliptic curves in class 82368en do not have complex multiplication.Modular form 82368.2.a.en
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
