# Properties

 Label 26b2 Conductor $26$ Discriminant $-125497034$ j-invariant $$-\frac{1064019559329}{125497034}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-213x-1257$$ y^2+xy+y=x^3-x^2-213x-1257 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-213xz^2-1257z^3$$ y^2z+xyz+yz^2=x^3-x^2z-213xz^2-1257z^3 (dehomogenize, simplify) $$y^2=x^3-3403x-83834$$ y^2=x^3-3403x-83834 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -213, -1257])

gp: E = ellinit([1, -1, 1, -213, -1257])

magma: E := EllipticCurve([1, -1, 1, -213, -1257]);

oscar: E = EllipticCurve([1, -1, 1, -213, -1257])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

trivial

magma: MordellWeilGroup(E);

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$26$$ = $2 \cdot 13$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-125497034$ = $-1 \cdot 2 \cdot 13^{7}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{1064019559329}{125497034}$$ = $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 13^{-7} \cdot 41^{3} \cdot 83^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.28979452610338456519600707002\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.28979452610338456519600707002\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.62096534954905546637586267270\dots$ comment: Real Period  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); Tamagawa product: $1$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $1$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $0.62096534954905546637586267270$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 0.620965350 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.620965 \cdot 1.000000 \cdot 1}{1^2} \approx 0.620965350$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 14
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is semistable. There are 2 primes of bad reduction:

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

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

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

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

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
$7$ 7B.1.3 7.48.0.5

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[8, 5, 91, 57], [185, 528, 350, 561], [1, 14, 0, 1], [1, 0, 14, 1], [120, 7, 441, 722], [8, 7, 357, 722], [715, 14, 714, 15], [8, 7, 175, 722]]

GL(2,Integers(728)).subgroup(gens)

Gens := [[8, 5, 91, 57], [185, 528, 350, 561], [1, 14, 0, 1], [1, 0, 14, 1], [120, 7, 441, 722], [8, 7, 357, 722], [715, 14, 714, 15], [8, 7, 175, 722]];

sub<GL(2,Integers(728))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$728 = 2^{3} \cdot 7 \cdot 13$$, index $96$, genus $2$, and generators

$\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 185 & 528 \\ 350 & 561 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 120 & 7 \\ 441 & 722 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 357 & 722 \end{array}\right),\left(\begin{array}{rr} 715 & 14 \\ 714 & 15 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 175 & 722 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[728])$ is a degree-$845365248$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/728\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 26b consists of 2 curves linked by isogenies of degree 7.

## Twists

This elliptic curve is its own minimal quadratic twist.

## 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 $K$ $3$ 3.1.104.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1124864.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ $$\Q(\zeta_{7})$$ $$\Z/7\Z$$ Not in database $7$ 7.1.52706752.1 $$\Z/7\Z$$ Not in database $8$ 8.2.999406512.1 $$\Z/3\Z$$ Not in database $12$ 12.2.8421963387109376.7 $$\Z/4\Z$$ Not in database $18$ 18.0.6007179870010464504905728.1 $$\Z/14\Z$$ Not in database $21$ 21.1.301059380309170415238823998755176448.1 $$\Z/14\Z$$ Not in database

We only show fields where the torsion growth is primitive.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 13 split ss ord ord nonsplit 1 0,0 0 4 0 0 0,0 0 1 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

## $p$-adic regulators

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