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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 29760b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29760.n6 | 29760b1 | \([0, -1, 0, 3839, -519935]\) | \(23862997439/457113600\) | \(-119829587558400\) | \([2]\) | \(98304\) | \(1.3818\) | \(\Gamma_0(N)\)-optimal |
| 29760.n5 | 29760b2 | \([0, -1, 0, -78081, -7909119]\) | \(200828550012481/12454560000\) | \(3264888176640000\) | \([2, 2]\) | \(196608\) | \(1.7284\) | |
| 29760.n4 | 29760b3 | \([0, -1, 0, -236801, 34659585]\) | \(5601911201812801/1271193750000\) | \(333235814400000000\) | \([2]\) | \(393216\) | \(2.0750\) | |
| 29760.n2 | 29760b4 | \([0, -1, 0, -1230081, -524696319]\) | \(785209010066844481/3324675600\) | \(871543760486400\) | \([2, 2]\) | \(393216\) | \(2.0750\) | |
| 29760.n3 | 29760b5 | \([0, -1, 0, -1210881, -541887999]\) | \(-749011598724977281/51173462246460\) | \(-13414816087136010240\) | \([2]\) | \(786432\) | \(2.4215\) | |
| 29760.n1 | 29760b6 | \([0, -1, 0, -19681281, -33600317439]\) | \(3216206300355197383681/57660\) | \(15115223040\) | \([2]\) | \(786432\) | \(2.4215\) |
Rank
sage: E.rank()
The elliptic curves in class 29760b have rank \(1\).
Complex multiplication
The elliptic curves in class 29760b do not have complex multiplication.Modular form 29760.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
