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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 19320r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19320.k4 | 19320r1 | \([0, -1, 0, -92575, 10872400]\) | \(5483900709072173056/277725\) | \(4443600\) | \([4]\) | \(40960\) | \(1.1971\) | \(\Gamma_0(N)\)-optimal |
| 19320.k3 | 19320r2 | \([0, -1, 0, -92580, 10871172]\) | \(342799332162880336/77131175625\) | \(19745580960000\) | \([2, 4]\) | \(81920\) | \(1.5437\) | |
| 19320.k2 | 19320r3 | \([0, -1, 0, -103160, 8243100]\) | \(118566490663726564/40187675390625\) | \(41152179600000000\) | \([2, 2]\) | \(163840\) | \(1.8902\) | |
| 19320.k5 | 19320r4 | \([0, -1, 0, -82080, 13420572]\) | \(-59722927783102084/41113267272525\) | \(-42099985687065600\) | \([4]\) | \(163840\) | \(1.8902\) | |
| 19320.k1 | 19320r5 | \([0, -1, 0, -678160, -208646900]\) | \(16841893263968213282/543703603314375\) | \(1113504979587840000\) | \([2]\) | \(327680\) | \(2.2368\) | |
| 19320.k6 | 19320r6 | \([0, -1, 0, 302560, 56767212]\) | \(1495639267637215678/1547698974609375\) | \(-3169687500000000000\) | \([2]\) | \(327680\) | \(2.2368\) |
Rank
sage: E.rank()
The elliptic curves in class 19320r have rank \(0\).
Complex multiplication
The elliptic curves in class 19320r do not have complex multiplication.Modular form 19320.2.a.r
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.
