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SageMath
E = EllipticCurve("di1")
E.isogeny_class()
Elliptic curves in class 111090.di
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111090.di1 | 111090dd8 | \([1, 0, 0, -3412590, -1526860650]\) | \(29689921233686449/10380965400750\) | \(1536755441778267516750\) | \([2]\) | \(7299072\) | \(2.7668\) | |
| 111090.di2 | 111090dd5 | \([1, 0, 0, -3047580, -2048021928]\) | \(21145699168383889/2593080\) | \(383868903048120\) | \([2]\) | \(2433024\) | \(2.2175\) | |
| 111090.di3 | 111090dd6 | \([1, 0, 0, -1428840, 639791100]\) | \(2179252305146449/66177562500\) | \(9796654296540562500\) | \([2, 2]\) | \(3649536\) | \(2.4202\) | |
| 111090.di4 | 111090dd3 | \([1, 0, 0, -1418260, 649983872]\) | \(2131200347946769/2058000\) | \(304657859562000\) | \([2]\) | \(1824768\) | \(2.0736\) | |
| 111090.di5 | 111090dd2 | \([1, 0, 0, -190980, -31833648]\) | \(5203798902289/57153600\) | \(8460783985550400\) | \([2, 2]\) | \(1216512\) | \(1.8709\) | |
| 111090.di6 | 111090dd4 | \([1, 0, 0, -42860, -79913400]\) | \(-58818484369/18600435000\) | \(-2753531931011715000\) | \([2]\) | \(2433024\) | \(2.2175\) | |
| 111090.di7 | 111090dd1 | \([1, 0, 0, -21700, 431120]\) | \(7633736209/3870720\) | \(573005476270080\) | \([2]\) | \(608256\) | \(1.5243\) | \(\Gamma_0(N)\)-optimal |
| 111090.di8 | 111090dd7 | \([1, 0, 0, 385630, 2154147762]\) | \(42841933504271/13565917968750\) | \(-2008242726604980468750\) | \([2]\) | \(7299072\) | \(2.7668\) |
Rank
sage: E.rank()
The elliptic curves in class 111090.di have rank \(1\).
Complex multiplication
The elliptic curves in class 111090.di do not have complex multiplication.Modular form 111090.2.a.di
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
