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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 54978bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54978.be4 | 54978bg1 | \([1, 0, 1, -326660, -54088702]\) | \(32765849647039657/8229948198912\) | \(968245175653797888\) | \([2]\) | \(1032192\) | \(2.1611\) | \(\Gamma_0(N)\)-optimal |
| 54978.be2 | 54978bg2 | \([1, 0, 1, -4858180, -4121581054]\) | \(107784459654566688937/10704361149504\) | \(1259357384877996096\) | \([2, 2]\) | \(2064384\) | \(2.5077\) | |
| 54978.be3 | 54978bg3 | \([1, 0, 1, -4491660, -4769588414]\) | \(-85183593440646799657/34223681512621656\) | \(-4026381906278425206744\) | \([2]\) | \(4128768\) | \(2.8543\) | |
| 54978.be1 | 54978bg4 | \([1, 0, 1, -77729020, -263774958142]\) | \(441453577446719855661097/4354701912\) | \(512326325244888\) | \([2]\) | \(4128768\) | \(2.8543\) |
Rank
sage: E.rank()
The elliptic curves in class 54978bg have rank \(1\).
Complex multiplication
The elliptic curves in class 54978bg do not have complex multiplication.Modular form 54978.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
