Show commands:
SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 10830h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10830.h4 | 10830h1 | \([1, 1, 0, -3617, -144411]\) | \(-111284641/123120\) | \(-5792288868720\) | \([2]\) | \(34560\) | \(1.1436\) | \(\Gamma_0(N)\)-optimal |
| 10830.h3 | 10830h2 | \([1, 1, 0, -68597, -6941319]\) | \(758800078561/324900\) | \(15285206736900\) | \([2, 2]\) | \(69120\) | \(1.4902\) | |
| 10830.h1 | 10830h3 | \([1, 1, 0, -1097447, -442967949]\) | \(3107086841064961/570\) | \(26816152170\) | \([2]\) | \(138240\) | \(1.8367\) | |
| 10830.h2 | 10830h4 | \([1, 1, 0, -79427, -4617201]\) | \(1177918188481/488703750\) | \(22991498466753750\) | \([2]\) | \(138240\) | \(1.8367\) |
Rank
sage: E.rank()
The elliptic curves in class 10830h have rank \(1\).
Complex multiplication
The elliptic curves in class 10830h do not have complex multiplication.Modular form 10830.2.a.h
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.
