Properties

Label 154.c1
Conductor $154$
Discriminant $2324168$
j-invariant \( \frac{15226621995131793}{2324168} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-x^2-5164x-141529\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-x^2z-5164xz^2-141529z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-82619x-9140458\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 1, -5164, -141529])
 
gp: E = ellinit([1, -1, 1, -5164, -141529])
 
magma: E := EllipticCurve([1, -1, 1, -5164, -141529]);
 
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);
 

\( \left(-\frac{165}{4}, \frac{161}{8}\right) \) Copy content Toggle raw display

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 154 \)  =  $2 \cdot 7 \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $2324168 $  =  $2^{3} \cdot 7^{4} \cdot 11^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{15226621995131793}{2324168} \)  =  $2^{-3} \cdot 3^{3} \cdot 7^{-4} \cdot 11^{-2} \cdot 82619^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.62717307818662955611022329097\dots$
Stable Faltings height: $0.62717307818662955611022329097\dots$

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: $0.56319601187170233145504220529\dots$
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: $ 12 $  = $ 3\cdot2\cdot2 $
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) $ ≈ $ 1.6895880356151069943651266159 $

Modular invariants

Modular form   154.2.a.c

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 96
$ \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$(j)_{-}$
$2$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

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 8.24.0.58

The image of the adelic Galois representation has level $56$, index $48$, and genus $0$.

$p$-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 7 11
Reduction type split ss nonsplit nonsplit
$\lambda$-invariant(s) 1 2,2 0 0
$\mu$-invariant(s) 1 0,0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 154.c 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{-1}) \) \(\Z/4\Z\) 2.0.4.1-11858.1-c4
$2$ \(\Q(\sqrt{-2}) \) \(\Z/4\Z\) 2.0.8.1-11858.2-c4
$4$ \(\Q(\zeta_{8})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ 4.2.247808.3 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.245635219456.5 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.143986855936.8 \(\Z/8\Z\) Not in database
$8$ 8.0.10070523904.5 \(\Z/8\Z\) Not in database
$8$ 8.2.76879700667.1 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ 16.0.1622647227216566419456.8 \(\Z/2\Z \oplus \Z/8\Z\) 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.