Properties

Label 78.a1
Conductor $78$
Discriminant $111045168$
j-invariant \( \frac{986551739719628473}{111045168} \)
CM no
Rank $0$
Torsion structure \(\Z/{4}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3+x^2-20739x+1140957\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z-20739xz^2+1140957z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-26878419x+53635662702\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, 1, 0, -20739, 1140957])
 
gp: E = ellinit([1, 1, 0, -20739, 1140957])
 
magma: E := EllipticCurve([1, 1, 0, -20739, 1140957]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

\(\Z/{4}\Z\)

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(86, 9\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(86, 9\right) \), \( \left(86, -95\right) \) Copy content Toggle raw display

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: $111045168 $  =  $2^{4} \cdot 3^{5} \cdot 13^{4} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{986551739719628473}{111045168} \)  =  $2^{-4} \cdot 3^{-5} \cdot 13^{-4} \cdot 109^{3} \cdot 9133^{3}$
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: $1.4504359261765928469744084519\dots$
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: $ 8 $  = $ 2\cdot1\cdot2^{2} $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
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

Modular form   78.2.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);
 

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

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

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
$3$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5
$13$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

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 4.12.0.7
sage: gens = [[7, 6, 306, 307], [1, 0, 8, 1], [43, 42, 130, 283], [40, 203, 43, 72], [112, 3, 109, 2], [145, 8, 268, 33], [1, 8, 0, 1], [1, 4, 4, 17], [305, 8, 304, 9]]
 
sage: GL(2,Integers(312)).subgroup(gens)
 
magma: Gens := [[7, 6, 306, 307], [1, 0, 8, 1], [43, 42, 130, 283], [40, 203, 43, 72], [112, 3, 109, 2], [145, 8, 268, 33], [1, 8, 0, 1], [1, 4, 4, 17], [305, 8, 304, 9]];
 
magma: sub<GL(2,Integers(312))|Gens>;
 

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

$\left(\begin{array}{rr} 7 & 6 \\ 306 & 307 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 43 & 42 \\ 130 & 283 \end{array}\right),\left(\begin{array}{rr} 40 & 203 \\ 43 & 72 \end{array}\right),\left(\begin{array}{rr} 112 & 3 \\ 109 & 2 \end{array}\right),\left(\begin{array}{rr} 145 & 8 \\ 268 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 305 & 8 \\ 304 & 9 \end{array}\right)$

$p$-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

$p$ 2 3 13
Reduction type nonsplit nonsplit split
$\lambda$-invariant(s) 0 0 1
$\mu$-invariant(s) 0 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 78.a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{3}) \) \(\Z/2\Z \oplus \Z/4\Z\) 2.2.12.1-1014.1-e5
$4$ 4.4.8112.1 \(\Z/8\Z\) Not in database
$8$ 8.0.47775744.1 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ 8.8.9475854336.1 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.0.1364523024384.6 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.2.80951927472.1 \(\Z/12\Z\) Not in database
$16$ deg 16 \(\Z/16\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/12\Z\) 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.