Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -421824000385, 105449485114502783]); // or

magma: E := EllipticCurve("212160z7");

sage: E = EllipticCurve([0, 1, 0, -421824000385, 105449485114502783]) # or

sage: E = EllipticCurve("212160z7")

gp: E = ellinit([0, 1, 0, -421824000385, 105449485114502783]) \\ or

gp: E = ellinit("212160z7")

$$y^2 = x^{3} + x^{2} - 421824000385 x + 105449485114502783$$

## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(\frac{16383462866}{43681}, -\frac{907238878875}{9129329}\right)$$ $$\hat{h}(P)$$ ≈ 11.89126991

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(374977, 0\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(374977, 0\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$212160$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$27057532649472000000$$ = $$2^{20} \cdot 3^{2} \cdot 5^{6} \cdot 13^{3} \cdot 17^{4}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{31664865542564944883878115208137569}{103216295812500}$$ = $$2^{-2} \cdot 3^{-2} \cdot 5^{-6} \cdot 11^{3} \cdot 13^{-3} \cdot 17^{-4} \cdot 433^{3} \cdot 66422003^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$11.89126991$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$0.04447616206$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$96$$  = $$2\cdot2\cdot( 2 \cdot 3 )\cdot1\cdot2^{2}$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 212160.2.a.hq

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q + q^{3} + q^{5} + 4q^{7} + q^{9} - q^{13} + q^{15} + q^{17} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 637009920 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

$$L'(E,1)$$ ≈ $$12.6930731427$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_10^{*}$$ Additive -1 6 20 2
$$3$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$5$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$13$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$17$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13e.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B
$$3$$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 212160.hq consists of 8 curves linked by isogenies of degrees dividing 12.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{6})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{26})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{2})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{13})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 $$\Q(\sqrt{2}, \sqrt{3})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{2}, \sqrt{13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{6}, \sqrt{13})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
$$\Q(\sqrt{6}, \sqrt{26})$$ $$\Z/12\Z$$ Not in database
6 6.0.10100390971392.17 $$\Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.