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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 11310j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11310.j4 | 11310j1 | \([1, 1, 1, -10335, 400005]\) | \(122083727651299441/32242728960\) | \(32242728960\) | \([4]\) | \(20480\) | \(1.0008\) | \(\Gamma_0(N)\)-optimal |
11310.j3 | 11310j2 | \([1, 1, 1, -11615, 292997]\) | \(173294065906331761/61964605497600\) | \(61964605497600\) | \([2, 4]\) | \(40960\) | \(1.3474\) | |
11310.j2 | 11310j3 | \([1, 1, 1, -78895, -8345755]\) | \(54309086480107021681/1575939143610000\) | \(1575939143610000\) | \([2, 2]\) | \(81920\) | \(1.6939\) | |
11310.j6 | 11310j4 | \([1, 1, 1, 35185, 2108837]\) | \(4817210305461175439/4682306425314960\) | \(-4682306425314960\) | \([4]\) | \(81920\) | \(1.6939\) | |
11310.j1 | 11310j5 | \([1, 1, 1, -1253395, -540629155]\) | \(217764763259392950709681/191615146362900\) | \(191615146362900\) | \([2]\) | \(163840\) | \(2.0405\) | |
11310.j5 | 11310j6 | \([1, 1, 1, 19125, -27596883]\) | \(773618103830753999/329643718157812500\) | \(-329643718157812500\) | \([2]\) | \(163840\) | \(2.0405\) |
Rank
sage: E.rank()
The elliptic curves in class 11310j have rank \(1\).
Complex multiplication
The elliptic curves in class 11310j do not have complex multiplication.Modular form 11310.2.a.j
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.