# Properties

 Label 77315b1 Conductor $77315$ Discriminant $541205$ j-invariant $$\frac{4879681}{245}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -46, 111])

gp: E = ellinit([1, 0, 0, -46, 111])

magma: E := EllipticCurve([1, 0, 0, -46, 111]);

$$y^2+xy=x^3-46x+111$$

## Mordell-Weil group structure

$\Z^2$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(5, 1\right)$$ $$\left(-2, 15\right)$$ $\hat{h}(P)$ ≈ $0.58325771381994873439902377630$ $1.1016238448865063632896726844$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2, 15\right)$$, $$\left(-2, -13\right)$$, $$\left(-1, 13\right)$$, $$\left(-1, -12\right)$$, $$\left(3, 0\right)$$, $$\left(3, -3\right)$$, $$\left(5, 1\right)$$, $$\left(5, -6\right)$$, $$\left(10, 21\right)$$, $$\left(10, -31\right)$$, $$\left(705, 18369\right)$$, $$\left(705, -19074\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$77315$$ = $5 \cdot 7 \cdot 47^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $541205$ = $5 \cdot 7^{2} \cdot 47^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4879681}{245}$$ = $5^{-1} \cdot 7^{-2} \cdot 47^{4}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.14125754570583812673923766783\dots$ Stable Faltings height: $-0.78294881265751455787606277946\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.62027018645378349975157151029\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $2.8856377174248591929755077919\dots$ 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: $2$  = $1\cdot2\cdot1$ 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: $1$ (rounded) 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); Special value: $L^{(2)}(E,1)/2!$ ≈ $3.5797500900503752701578341662492479280$

## Modular invariants

Modular form 77315.2.a.a

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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 12288 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

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$(j)_{-}$
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$47$ $1$ $II$ Additive -1 2 2 0

## Galois representations

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$.

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 ordinary ordinary nonsplit nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary add 3 2 6 2 2 2 2 2 2 2 2 2 2 2 - 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 77315b consists of this curve only.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.3.11045.1 $$\Z/2\Z$$ Not in database $6$ 6.6.609960125.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.