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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) 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 Real period: $$4.802421322$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$2$$  = $$2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form121.2.a.b

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

 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/factorial(ar)

$$L'(E,1)$$ ≈ $$0.86237229669$$

## Local data

This elliptic curve is not semistable.

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

sage: E.local_data()

gp: ellglobalred(E)

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

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

For the identification of this curve $E$ with the modular curve $X_{\text{nonsplit}}(11)$, see [E. Halberstadt's paper] "Sur la courbe modulaire $X_{ndép}(11)$" ["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 $\mathcal{E}$ 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 $\Q(\sqrt{\pm13})$.