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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 38115m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 38115.h6 | 38115m1 | \([1, -1, 1, 38092, -831265234]\) | \(4733169839/231139696095\) | \(-298509513871086881055\) | \([2]\) | \(921600\) | \(2.6077\) | \(\Gamma_0(N)\)-optimal |
| 38115.h5 | 38115m2 | \([1, -1, 1, -13035353, -17790138088]\) | \(189674274234120481/3859869269025\) | \(4984897525473231360225\) | \([2, 2]\) | \(1843200\) | \(2.9542\) | |
| 38115.h4 | 38115m3 | \([1, -1, 1, -27709628, 29613639872]\) | \(1821931919215868881/761147600816295\) | \(982997746135443245754855\) | \([2]\) | \(3686400\) | \(3.3008\) | |
| 38115.h2 | 38115m4 | \([1, -1, 1, -207536198, -1150718660044]\) | \(765458482133960722801/326869475625\) | \(422141457813513755625\) | \([2, 2]\) | \(3686400\) | \(3.3008\) | |
| 38115.h3 | 38115m5 | \([1, -1, 1, -206507093, -1162696207318]\) | \(-754127868744065783521/15825714261328125\) | \(-20438403055051768836328125\) | \([2]\) | \(7372800\) | \(3.6474\) | |
| 38115.h1 | 38115m6 | \([1, -1, 1, -3320578823, -73648500528094]\) | \(3135316978843283198764801/571725\) | \(738364524576525\) | \([2]\) | \(7372800\) | \(3.6474\) |
Rank
sage: E.rank()
The elliptic curves in class 38115m have rank \(1\).
Complex multiplication
The elliptic curves in class 38115m do not have complex multiplication.Modular form 38115.2.a.m
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.
