# Properties

 Label 42978.u1 Conductor 42978 Discriminant -5737242625602477531070464 j-invariant $$-\frac{10748395438529140294639078020769}{5737242625602477531070464}$$ CM no Rank 1 Torsion Structure $$\Z/{7}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -459769306, 3796241996132]) # or

sage: E = EllipticCurve("42978r1")

gp: E = ellinit([1, 0, 0, -459769306, 3796241996132]) \\ or

gp: E = ellinit("42978r1")

magma: E := EllipticCurve([1, 0, 0, -459769306, 3796241996132]); // or

magma: E := EllipticCurve("42978r1");

$$y^2 + x y = x^{3} - 459769306 x + 3796241996132$$

## Mordell-Weil group structure

$$\Z\times \Z/{7}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(9812, 474326\right)$$ $$\hat{h}(P)$$ ≈ 2.5752300457588895

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-2476, 2219222\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2476, 2219222\right)$$, $$\left(-2476, -2216746\right)$$, $$\left(9812, 474326\right)$$, $$\left(9812, -484138\right)$$, $$\left(10292, 387926\right)$$, $$\left(10292, -398218\right)$$, $$\left(12116, 59606\right)$$, $$\left(12116, -71722\right)$$, $$\left(12686, 66218\right)$$, $$\left(12686, -78904\right)$$, $$\left(14852, 486422\right)$$, $$\left(14852, -501274\right)$$, $$\left(34004, 5225174\right)$$, $$\left(34004, -5259178\right)$$, $$\left(40844, 7270334\right)$$, $$\left(40844, -7311178\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$42978$$ = $$2 \cdot 3 \cdot 13 \cdot 19 \cdot 29$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-5737242625602477531070464$$ = $$-1 \cdot 2^{28} \cdot 3^{7} \cdot 13 \cdot 19^{7} \cdot 29^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{10748395438529140294639078020769}{5737242625602477531070464}$$ = $$-1 \cdot 2^{-28} \cdot 3^{-7} \cdot 13^{-1} \cdot 19^{-7} \cdot 29^{-2} \cdot 83^{3} \cdot 313^{3} \cdot 547^{3} \cdot 1553^{3}$$ Endomorphism ring: $$\Z$$ (no 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: $$2.57523004576$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0749460599905$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$2744$$  = $$( 2^{2} \cdot 7 )\cdot7\cdot1\cdot7\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$7$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 42978.2.a.u

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - 2q^{11} + q^{12} - q^{13} + q^{14} - q^{15} + q^{16} + 4q^{17} + q^{18} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 11063808 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

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

$$L'(E,1)$$ ≈ $$10.8081873479$$

## Local data

This elliptic curve is semistable.

sage: E.local_data()

gp: ellglobalred(E)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$28$$ $$I_{28}$$ Split multiplicative -1 1 28 28
$$3$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$13$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$19$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$29$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ except those listed.

prime Image of Galois representation
$$7$$ B.1.1

## $p$-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 $$p\ge5$$ 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 split split ordinary ordinary ordinary nonsplit ordinary split ordinary split ordinary ordinary ss ordinary ordinary 4 2 7 51 1 1 1 2 1 2 1 1 1,1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0,0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 42978.u consists of 2 curves linked by isogenies of degree 7.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.2964.1 $$\Z/14\Z$$ Not in database
6 6.0.26039617344.2 $$\Z/2\Z \times \Z/14\Z$$ 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.