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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 2394n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2394.j6 | 2394n1 | \([1, -1, 1, -70205, -7142155]\) | \(52492168638015625/293197968\) | \(213741318672\) | \([2]\) | \(9216\) | \(1.3664\) | \(\Gamma_0(N)\)-optimal |
| 2394.j5 | 2394n2 | \([1, -1, 1, -71465, -6871507]\) | \(55369510069623625/3916046302812\) | \(2854797754749948\) | \([2]\) | \(18432\) | \(1.7129\) | |
| 2394.j4 | 2394n3 | \([1, -1, 1, -100580, -352249]\) | \(154357248921765625/89242711068672\) | \(65057936369061888\) | \([6]\) | \(27648\) | \(1.9157\) | |
| 2394.j3 | 2394n4 | \([1, -1, 1, -1088420, 435877895]\) | \(195607431345044517625/752875610010048\) | \(548846319697324992\) | \([6]\) | \(55296\) | \(2.2622\) | |
| 2394.j2 | 2394n5 | \([1, -1, 1, -5505035, 4972852379]\) | \(25309080274342544331625/191933498523648\) | \(139919520423739392\) | \([6]\) | \(82944\) | \(2.4650\) | |
| 2394.j1 | 2394n6 | \([1, -1, 1, -88080395, 318197707931]\) | \(103665426767620308239307625/5961940992\) | \(4346254983168\) | \([6]\) | \(165888\) | \(2.8115\) |
Rank
sage: E.rank()
The elliptic curves in class 2394n have rank \(1\).
Complex multiplication
The elliptic curves in class 2394n do not have complex multiplication.Modular form 2394.2.a.n
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.
