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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 12789f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12789.k5 | 12789f1 | \([1, -1, 0, -345753, 83869344]\) | \(-53297461115137/4513839183\) | \(-387134477543719143\) | \([2]\) | \(147456\) | \(2.1198\) | \(\Gamma_0(N)\)-optimal |
| 12789.k4 | 12789f2 | \([1, -1, 0, -5639958, 5156776575]\) | \(231331938231569617/1472026689\) | \(126250019124003369\) | \([2, 2]\) | \(294912\) | \(2.4664\) | |
| 12789.k3 | 12789f3 | \([1, -1, 0, -5748003, 4949006040]\) | \(244883173420511137/18418027974129\) | \(1579642815810532683609\) | \([2, 2]\) | \(589824\) | \(2.8129\) | |
| 12789.k1 | 12789f4 | \([1, -1, 0, -90239193, 329967079434]\) | \(947531277805646290177/38367\) | \(3290588764407\) | \([2]\) | \(589824\) | \(2.8129\) | |
| 12789.k2 | 12789f5 | \([1, -1, 0, -18728838, -25356051351]\) | \(8471112631466271697/1662662681263647\) | \(142600128703442381503287\) | \([2]\) | \(1179648\) | \(3.1595\) | |
| 12789.k6 | 12789f6 | \([1, -1, 0, 5504112, 21955452651]\) | \(215015459663151503/2552757445339983\) | \(-218940103940679868115943\) | \([2]\) | \(1179648\) | \(3.1595\) |
Rank
sage: E.rank()
The elliptic curves in class 12789f have rank \(1\).
Complex multiplication
The elliptic curves in class 12789f do not have complex multiplication.Modular form 12789.2.a.f
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.
