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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 35378p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 35378.k5 | 35378p1 | \([1, 0, 0, -9213, 688141]\) | \(-15625/28\) | \(-154977223905532\) | \([2]\) | \(114048\) | \(1.4131\) | \(\Gamma_0(N)\)-optimal |
| 35378.k4 | 35378p2 | \([1, 0, 0, -186103, 30865575]\) | \(128787625/98\) | \(542420283669362\) | \([2]\) | \(228096\) | \(1.7597\) | |
| 35378.k6 | 35378p3 | \([1, 0, 0, 79232, -14400576]\) | \(9938375/21952\) | \(-121502143541937088\) | \([2]\) | \(342144\) | \(1.9624\) | |
| 35378.k3 | 35378p4 | \([1, 0, 0, -628328, -157186184]\) | \(4956477625/941192\) | \(5209404404360552648\) | \([2]\) | \(684288\) | \(2.3090\) | |
| 35378.k2 | 35378p5 | \([1, 0, 0, -3016343, -2022438167]\) | \(-548347731625/1835008\) | \(-10156587345872945152\) | \([2]\) | \(1026432\) | \(2.5117\) | |
| 35378.k1 | 35378p6 | \([1, 0, 0, -48300183, -129206631191]\) | \(2251439055699625/25088\) | \(138859592619356672\) | \([2]\) | \(2052864\) | \(2.8583\) |
Rank
sage: E.rank()
The elliptic curves in class 35378p have rank \(0\).
Complex multiplication
The elliptic curves in class 35378p do not have complex multiplication.Modular form 35378.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
