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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 330330.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330330.r1 | 330330r4 | \([1, 1, 0, -6261095997, -190690737682419]\) | \(15322113990427033386386178481/7914219455142000\) | \(14020522532170816662000\) | \([2]\) | \(283115520\) | \(4.0170\) | |
| 330330.r2 | 330330r5 | \([1, 1, 0, -3912483577, 93043751254249]\) | \(3738739308040935679248658801/52481104495587955687500\) | \(92973477961308294365703187500\) | \([2]\) | \(566231040\) | \(4.3636\) | |
| 330330.r3 | 330330r3 | \([1, 1, 0, -471546077, -1670806183251]\) | \(6545457555147351153658801/3107302952660156250000\) | \(5504776726117579066406250000\) | \([2, 2]\) | \(283115520\) | \(4.0170\) | |
| 330330.r4 | 330330r2 | \([1, 1, 0, -391385997, -2978585824419]\) | \(3742686521561161723938481/2688412771044000000\) | \(4762687217083479684000000\) | \([2, 2]\) | \(141557760\) | \(3.6704\) | |
| 330330.r5 | 330330r1 | \([1, 1, 0, -19519117, -65901300131]\) | \(-464245965066884560561/790125654269952000\) | \(-1399755794204130435072000\) | \([2]\) | \(70778880\) | \(3.3238\) | \(\Gamma_0(N)\)-optimal |
| 330330.r6 | 330330r6 | \([1, 1, 0, 1686830143, -12685863384399]\) | \(299627492656734072806910479/212643384933471679687500\) | \(-376710727656126022338867187500\) | \([2]\) | \(566231040\) | \(4.3636\) |
Rank
sage: E.rank()
The elliptic curves in class 330330.r have rank \(1\).
Complex multiplication
The elliptic curves in class 330330.r do not have complex multiplication.Modular form 330330.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
