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SageMath
E = EllipticCurve("bv1")
E.isogeny_class()
Elliptic curves in class 85176bv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85176.m5 | 85176bv1 | \([0, 0, 0, -364026, -81251651]\) | \(94757435392/4179357\) | \(235297381899866832\) | \([2]\) | \(688128\) | \(2.0963\) | \(\Gamma_0(N)\)-optimal |
| 85176.m4 | 85176bv2 | \([0, 0, 0, -980031, 266298370]\) | \(115562131792/32867289\) | \(29606801436091885824\) | \([2, 2]\) | \(1376256\) | \(2.4429\) | |
| 85176.m6 | 85176bv3 | \([0, 0, 0, 2579109, 1756154374]\) | \(526556774012/674481717\) | \(-2430288213608773260288\) | \([2]\) | \(2752512\) | \(2.7895\) | |
| 85176.m2 | 85176bv4 | \([0, 0, 0, -14395251, 21019643710]\) | \(91557481657828/12595401\) | \(45383668414508196864\) | \([2, 2]\) | \(2752512\) | \(2.7895\) | |
| 85176.m3 | 85176bv5 | \([0, 0, 0, -13117611, 24902902726]\) | \(-34639400027234/17130345141\) | \(-123447900341597010536448\) | \([2]\) | \(5505024\) | \(3.1361\) | |
| 85176.m1 | 85176bv6 | \([0, 0, 0, -230316411, 1345350486454]\) | \(187491149065688834/3549\) | \(25575468252751872\) | \([2]\) | \(5505024\) | \(3.1361\) |
Rank
sage: E.rank()
The elliptic curves in class 85176bv have rank \(1\).
Complex multiplication
The elliptic curves in class 85176bv do not have complex multiplication.Modular form 85176.2.a.bv
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.
