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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 12495b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12495.a4 | 12495b1 | \([1, 1, 1, -191346, -32296146]\) | \(6585576176607121/187425\) | \(22050363825\) | \([2]\) | \(49152\) | \(1.4944\) | \(\Gamma_0(N)\)-optimal |
12495.a3 | 12495b2 | \([1, 1, 1, -191591, -32209612]\) | \(6610905152742241/35128130625\) | \(4132789439900625\) | \([2, 2]\) | \(98304\) | \(1.8410\) | |
12495.a2 | 12495b3 | \([1, 1, 1, -299636, 8069564]\) | \(25288177725059761/14387797265625\) | \(1692709960503515625\) | \([2, 2]\) | \(196608\) | \(2.1876\) | |
12495.a5 | 12495b4 | \([1, 1, 1, -87466, -66945712]\) | \(-629004249876241/16074715228425\) | \(-1891174171908972825\) | \([2]\) | \(196608\) | \(2.1876\) | |
12495.a1 | 12495b5 | \([1, 1, 1, -3515261, 2530405814]\) | \(40832710302042509761/91556816413125\) | \(10771567894187743125\) | \([2]\) | \(393216\) | \(2.5342\) | |
12495.a6 | 12495b6 | \([1, 1, 1, 1187269, 65761478]\) | \(1573196002879828319/926055908203125\) | \(-108949551544189453125\) | \([2]\) | \(393216\) | \(2.5342\) |
Rank
sage: E.rank()
The elliptic curves in class 12495b have rank \(0\).
Complex multiplication
The elliptic curves in class 12495b do not have complex multiplication.Modular form 12495.2.a.b
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.