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Properties

Label 26.b1
Conductor $26$
Discriminant $-125497034$
j-invariant \( -\frac{1064019559329}{125497034} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-x^2-213x-1257\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-x^2z-213xz^2-1257z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-3403x-83834\) Copy content Toggle raw display (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 = elliptic_curve([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);
 

Invariants

Conductor: $N$  =  \( 26 \) = $2 \cdot 13$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $\Delta$  =  $-125497034$ = $-1 \cdot 2 \cdot 13^{7} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: $j$  =  \( -\frac{1064019559329}{125497034} \) = $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 13^{-7} \cdot 41^{3} \cdot 83^{3}$
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: $\mathrm{End}(E)$ = $\Z$
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$  =  \(\Z\)    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$
Faltings height: $h_{\mathrm{Faltings}}$ ≈ $0.28979452610338456519600707002$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $h_{\mathrm{stable}}$ ≈ $0.28979452610338456519600707002$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $Q$ ≈ $1.0626891964834324$
Szpiro ratio: $\sigma_{m}$ ≈ $8.556703863672293$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$ = $ 0$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: $r$ = $ 0$
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: $\mathrm{Reg}(E/\Q)$ = $1$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $\Omega$ ≈ $0.62096534954905546637586267270$
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: $\prod_{p}c_p$ = $ 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: $\#E(\Q)_{\mathrm{tor}}$ = $1$
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
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);
 
Analytic order of Ш: Ш${}_{\mathrm{an}}$  =  $1$    (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

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 analytic 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

Modular form   26.2.a.b

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

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);
 

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

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 at primes of bad reduction

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

$p$ Tamagawa number Kodaira symbol Reduction type Root number $\mathrm{ord}_p(N)$ $\mathrm{ord}_p(\Delta)$ $\mathrm{ord}_p(\mathrm{den}(j))$
$2$ $1$ $I_{1}$ split multiplicative -1 1 1 1
$13$ $1$ $I_{7}$ nonsplit 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 $\GL_2(\Z/728\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$2$ split multiplicative $4$ \( 13 \)
$7$ good $2$ \( 2 \)
$13$ nonsplit multiplicative $14$ \( 2 \)

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 26.b 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 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

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