# Properties

 Label 78a3 Conductor $78$ Discriminant $725251155408$ j-invariant $$\frac{1416134368422073}{725251155408}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3+x^2-2339x-15747$$ y^2+xy=x^3+x^2-2339x-15747 (homogenize, simplify) $$y^2z+xyz=x^3+x^2z-2339xz^2-15747z^3$$ y^2z+xyz=x^3+x^2z-2339xz^2-15747z^3 (dehomogenize, simplify) $$y^2=x^3-3032019x-689215122$$ y^2=x^3-3032019x-689215122 (homogenize, minimize)

sage: E = EllipticCurve([1, 1, 0, -2339, -15747])

gp: E = ellinit([1, 1, 0, -2339, -15747])

magma: E := EllipticCurve([1, 1, 0, -2339, -15747]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{181}{4}, \frac{181}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$78$$ = $2 \cdot 3 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $725251155408$ = $2^{4} \cdot 3^{20} \cdot 13$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1416134368422073}{725251155408}$$ = $2^{-4} \cdot 3^{-20} \cdot 13^{-1} \cdot 112297^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.96807765231754098127344359430\dots$ Stable Faltings height: $0.96807765231754098127344359430\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.72521796308829642348720422595\dots$ 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: $4$  = $2\cdot2\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) 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: $L(E,1)$ ≈ $0.72521796308829642348720422595$

## Modular invariants

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 160 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is 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$(j)_{-}$
$2$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$3$ $2$ $I_{20}$ Non-split multiplicative 1 1 20 20
$13$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

## 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 for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.12.0.8
sage: gens = [[7, 6, 306, 307], [224, 3, 101, 2], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [203, 196, 242, 45], [43, 42, 130, 283], [305, 8, 304, 9], [209, 8, 212, 33]]

sage: GL(2,Integers(312)).subgroup(gens)

magma: Gens := [[7, 6, 306, 307], [224, 3, 101, 2], [1, 0, 8, 1], [1, 8, 0, 1], [1, 4, 4, 17], [203, 196, 242, 45], [43, 42, 130, 283], [305, 8, 304, 9], [209, 8, 212, 33]];

magma: sub<GL(2,Integers(312))|Gens>;

The image of the adelic Galois representation has level $312$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 7 & 6 \\ 306 & 307 \end{array}\right),\left(\begin{array}{rr} 224 & 3 \\ 101 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 203 & 196 \\ 242 & 45 \end{array}\right),\left(\begin{array}{rr} 43 & 42 \\ 130 & 283 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 304 & 9 \end{array}\right),\left(\begin{array}{rr} 209 & 8 \\ 212 & 33 \end{array}\right)$

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 13 nonsplit nonsplit split 0 0 1 1 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 78a consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{13})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.2.13.1-468.1-e3 $2$ $$\Q(\sqrt{-13})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ 2.0.4.1-3042.2-a2 $4$ 4.2.35152.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{13})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.19770609664.3 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.25622710124544.30 $$\Z/8\Z$$ Not in database $8$ 8.0.897122304.10 $$\Z/8\Z$$ Not in database $8$ 8.2.80951927472.1 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ 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.