Properties

Label 121.b2
Conductor 121
Discriminant -1331
j-invariant \( -32768 \)
CM yes (\(D=-11\))
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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This is the modular curve $X_{\text{nonsplit}}(11)$.

Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 1, -7, 10]); // or
 
magma: E := EllipticCurve("121b1");
 
sage: E = EllipticCurve([0, -1, 1, -7, 10]) # or
 
sage: E = EllipticCurve("121b1")
 
gp: E = ellinit([0, -1, 1, -7, 10]) \\ or
 
gp: E = ellinit("121b1")
 

\( y^2 + y = x^{3} - x^{2} - 7 x + 10 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
 
sage: E.gens()
 

\(P\) =  \( \left(4, 5\right) \)
\(\hat{h}(P)\) ≈  0.0897851561607

Integral points

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

\( \left(-2, 3\right) \), \( \left(-2, -4\right) \), \( \left(2, 0\right) \), \( \left(2, -1\right) \), \( \left(4, 5\right) \), \( \left(4, -6\right) \)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 121 \)  =  \(11^{2}\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-1331 \)  =  \(-1 \cdot 11^{3} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -32768 \)  =  \(-1 \cdot 2^{15}\)
Endomorphism ring: \(\Z[(1+\sqrt{-11})/2]\)   ( Complex Multiplication)
Sato-Tate Group: $N(\mathrm{U}(1))$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.0897851561607\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(4.802421322\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 2 \)  = \( 2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 121.2.a.b

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q - q^{3} - 2q^{4} - 3q^{5} - 2q^{9} + 2q^{12} + 3q^{15} + 4q^{16} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 4
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L'(E,1) \) ≈ \( 0.86237229669 \)

Local data

This elliptic curve is not semistable.

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(11\) \(2\) \( III \) Additive 1 2 3 0

Galois representations

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois representation
\(11\) B.1.3

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=-1\).

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ordinary ordinary ss add ss ss ss ordinary ss ordinary ordinary ss ss ordinary
$\lambda$-invariant(s) ? 1 1 1,1 - 1,1 1,1 1,1 1 1,1 1 1 1,1 1,1 1
$\mu$-invariant(s) ? 0 0 0,0 - 0,0 0,0 0,0 0 0,0 0 0 0,0 0,0 0

An entry ? indicates that the invariants have not yet been computed.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 11.
Its isogeny class 121.b consists of 2 curves linked by isogenies of degree 11.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 3.1.44.1 \(\Z/2\Z\) Not in database
4 4.0.3993.1 \(\Z/3\Z\) Not in database
4.2.11979.1 \(\Z/3\Z\) Not in database
5 \(\Q(\zeta_{11})^+\) \(\Z/11\Z\) 5.5.14641.1-121.1-b1
6 6.0.21296.1 \(\Z/2\Z \times \Z/2\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.

Additional information

For the identification of this curve $E$ with the modular curve $X_{\text{nonsplit}}(11)$, see [E. Halberstadt's paper] "Sur la courbe modulaire $X_{nd├ęp}(11)$" ["On the modular curve $X_{\text{nonsplit}}(11)$"], Experimental Math. 7 (1998) #2, 163-174. This modular curve classifies (up to twist) elliptic curves $\mathcal E$ such that the Galois action on $\mathcal{E}[11]$ factors through the normalizer of a non-split Cartan subgroup of $\text{GL}_2({\bf F}_{11})$. Since $E(\Q)$ is infinite, there are infinitely many $j$-invariants of such $\mathcal E$. Halberstadt computes the first few examples; most are CM curves, and of the non-CM curves the first conductor (and the oniy one within the LMFDB range $N \leq 4\cdot 10^5$) is $232544 = 2^5 13^2 43$, arising for the curves [232544.f1], [232544.g1], [232544.h1], and [232544.i1] which are related by quadratic twists by $\Q(\sqrt{-1})$ and $\Q(\sqrt{\pm13})$.