# Properties

 Label 69366.u1 Conductor 69366 Discriminant -373958095272087819200664 j-invariant $$-\frac{925492188434597796818942768449}{373958095272087819200664}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

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Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -203025076, -1113860484568]) # or

sage: E = EllipticCurve("69366w2")

gp: E = ellinit([1, 0, 0, -203025076, -1113860484568]) \\ or

gp: E = ellinit("69366w2")

magma: E := EllipticCurve([1, 0, 0, -203025076, -1113860484568]); // or

magma: E := EllipticCurve("69366w2");

$$y^2 + x y = x^{3} - 203025076 x - 1113860484568$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{24880589116471}{298944100}, -\frac{122366764350732057239}{5168743489000}\right)$$ $$\hat{h}(P)$$ ≈ 30.93937299196835

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$69366$$ = $$2 \cdot 3 \cdot 11 \cdot 1051$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-373958095272087819200664$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 11 \cdot 1051^{7}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{925492188434597796818942768449}{373958095272087819200664}$$ = $$-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 11^{-1} \cdot 43^{3} \cdot 97^{3} \cdot 127^{3} \cdot 1051^{-7} \cdot 18397^{3}$$ 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: $$30.939372992$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0199973115272$$ 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: $$21$$  = $$3\cdot1\cdot1\cdot7$$ sage: E.torsion_order()  gp: elltors(E)[1]  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 69366.2.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);

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

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 13253520 $$\Gamma_0(N)$$-optimal: no 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);

$$L'(E,1)$$ ≈ $$12.9927898837$$

## 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$$(j)_{-}$$
$$2$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$3$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$11$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$1051$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7

## 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.1.3

## $p$-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.

## 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 1051 split split ordinary ordinary split ss ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary split 3 4 1 3 4 1,1 5 1 1 1 1 1 1,1 1 1 ? 0 0 0 1 0 0,0 0 0 0 0 0 0 0,0 0 0 ?

An entry ? indicates that the invariants have not yet been computed.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 69366.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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.277464.4 $$\Z/2\Z$$ Not in database
6 $$\Q(\zeta_{7})$$ $$\Z/7\Z$$ Not in database
$$x^{6} - 194 x^{4} + 9409 x^{2} + 1109856$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
7 7.1.68069081958026688.1 $$\Z/7\Z$$ 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.