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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -7, -49]) # or

sage: E = EllipticCurve("25270h1")

gp: E = ellinit([1, 1, 0, -7, -49]) \\ or

gp: E = ellinit("25270h1")

magma: E := EllipticCurve([1, 1, 0, -7, -49]); // or

magma: E := EllipticCurve("25270h1");

$$y^2 + x y = x^{3} + x^{2} - 7 x - 49$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(7, -21\right)$$ $$\hat{h}(P)$$ ≈ $0.5194066299850294$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(5, 6\right)$$, $$\left(5, -11\right)$$, $$\left(7, 14\right)$$, $$\left(7, -21\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$25270$$ = $$2 \cdot 5 \cdot 7 \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-884450$$ = $$-1 \cdot 2 \cdot 5^{2} \cdot 7^{2} \cdot 19^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{130321}{2450}$$ = $$-1 \cdot 2^{-1} \cdot 5^{-2} \cdot 7^{-2} \cdot 19^{4}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.519406629985$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$1.20835839789$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$4$$  = $$1\cdot2\cdot2\cdot1$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

Modular form 25270.2.a.c

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} - 2q^{9} - q^{10} - 5q^{11} - q^{12} + q^{14} - q^{15} + q^{16} + 2q^{17} + 2q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3744 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

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

gp: ar = ellanalyticrank(E);

gp: ar/factorial(ar)

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

$$L'(E,1)$$ ≈ $$2.51051745305$$

## Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$19$$ $$1$$ $$II$$ Additive -1 2 2 0

## 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 X4.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.

sage: rho = E.galois_representation();

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

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

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

## $p$-adic data

### $p$-adic regulators

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

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ of good ordinary reduction.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 nonsplit ordinary split nonsplit ordinary ss ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary 7 1 2 1 1 1,1 1 - 1 1 1 1 1 1 1 0 0 0 0 0 0,0 0 - 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has no rational isogenies. Its isogeny class 25270h consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.2888.1 $$\Z/2\Z$$ Not in database
$6$ 6.0.66724352.2 $$\Z/2\Z \times \Z/2\Z$$ 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.